a new non-cubic equation of state
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Graduate Studies Legacy Theses
2000
A new non-cubic equation of state
Kedge, Christopher J.
Kedge, C. J. (2000). A new non-cubic equation of state (Unpublished master's thesis). University
of Calgary, Calgary, AB. doi:10.11575/PRISM/20257
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THE UNIVERSITY OF CALGARY
A NEW NON-CUBIC EQUATION OF STATE
Christopher J. Kedge
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES
[N PARTIAL FCILFILLMENT OF THE REQURBENTS FOR THE
DEGREE OF MASTER OF ENGINEERING
DEPARTMENT OF CHEMICAL AND PETROLEUM ENGINEERING
CALGARY, ALBERTA
JANUARY, 2000
0 Christopher l. Kedge 2000
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ABSTRACT
A new non-cubic equation of state (EOS) has been developed for pure substances. The
density dependence of the equation was devised by fitting critical isotherms of test
substances while reproducing the critical point, and enforcing van der Wads' conditions
of criticality. Temperature dependence was determined for three of the parameters by
forcing reproduction of saturation pressures and saturated Liquid and vapour specific
volumes at subcritical temperatures. The EOS was completed by developing Functions to
fit the calculated temperature dependence. A van der Waals-type covolume parameter
was included to provide a non-zero density limit, and to avoid generating spurious
volume roots.
The new equation was compared to the BWRS equation of state (Starling, 1973) and the
Soave-BWR equation (Soave, 1995). The new equation generally showed improvement
over the BWRS equation for fitting pure-substance critical isotherms, and for cdculating
saturation- md other thermodynamic properties, but performed better than the Soave-
B WR EOS in only a few areas.
ACKNOWLEDGMENTS
I would Like to tbauk Dr. Mark Trebble for his guidance and encouragement during the
years I've worked on this thesis. I also thank Bernie Moore and Dr. Janusz Stuchly for
friendship and inspiration during that same period. 1 thank Dr. PR. Bishnoi, Dr. AX.
Mehrotra, and Dr. W.J.D. Shaw for helphl suggestions. I thank my parents for Lifelong
love and support. And I thank my wife, Carolynne, for all these things-but above all,
for patience that has yet to run out.
Dedicated to my wife, Carolynne.
TABLE OF CONTENTS
Page
Approval Page Abstract Aclmowledgements Dedication Table of Contents List of Tables List of Figures List of Symbols
1.0 INTRODUCTION
2.0 NON-CUBIC EQUATIONS OF STATE 2.1 The Virial Equation of State 2.2 B WR-Type Equations of State 2.3 Repulsive Terms 2.4 Recent Developments
3.0 DEVELOPMENT OF THE NEW EQUATION OF STATE 3.1 Objectives and General Approach 3.2 Density Dependence: Fitting the Critical Isotherm
3 2.1 Initial Approach 3 2.2 Regression Procedure 3 2.3 EOS Development 3 2.4 Alternate Repulsive Terms 3 2.5 Additional Specifications: Second ViriaI
Coefficients and Third Derivatives 3 2.6 Sensitivity Analysis
3.3 Temperature Dependence of EOS Parameters 3.3.1 General Approach 3.3.2 Determining Temperature Dependence 3 -3.3 Temperature Function Development 3.3.4 Temperature Function Regression 3.3.5 Spurious Roots
3.4 Thermodynamic Properties fiom the New EOS 3 -4.1 Fugacity 3 -4.2 Enthdpy Departure
4.0 COMPARISON WITH PREVIOUSLY PUBLISHED NON-CUBIC EOS 4.1 Pure-Component Critical Isotherms 4.2 Saturation Properties 4.3 Single-Phase PVT 4-4 Enthalpy of Vaporization 4.5 Comparing Exponential Terms
vi viii ix
xii
5.0 DISCUSSION 5.1 Fitting the Critical Isotherm 5.2 The CovoIurne Parameter 5.3 Temperature Dependence 5.4 Further Development
6.0 CONCLUSIONS
7.0 REFERENCES
APPENDIX: EOS parameters for pure components
LIST OF TABLES
Values of 'b' parameter for alternate versions of the new EOS 35
Comparison of erron for two sets of temperature hc t ioos 55
Comparison of errors between predicted and experimental pressures 79 and specific volumes along pure substance criticai isotherms
Comparison of errors between predicted and smoothed experimental 80 saturation properties of methane, n-pentane, and sulfur dioxide
Comparison of errors between predicted and experimental pressures 86 dong supercritical isotherms of methane, pent- and sulfur dioxide
Error in vaporization enthdpy for methane 87
LIST OF FIGURES
3-1 Critical isotherm of methane
3-2 Error in pressure along CI critical isotherm for Eq. (16)
3-3 Error in pressure along CI critical isotherm for Eq. (1 7)
3 4 Error in pressure along Cl critical isotherm for Eq. (18)
3 -5 Error in pressure along CI critical isotherm for Eq. (20)
3 -6 Critical isotherm of Cl predicted by Eq. (20)
3-7 &or in pressure dong CI critical isotherm for alternate versions of the new EOS
3 -8 Error in specific volume along CI critical isotherm for alternate versions of the new EOS
3-9 Error in pressure dong CI critical isotherm for new EOS with forced second viriai coefficient
3- 10 Error in pressure along CI critical isotherm for new EOS with third derivative forced to zero at critical point
3 - 1 1 (a) Sensitivity of fit along C I critical isotherm to I % perturbation in Bo CEq. (2 111
(b) Sensitivity of fit along CI critical isotherm to 1 % perturbation in co CEq- (21 11
(c) Sensitivity of fit dong Ci critical isotherm to I % perturbation in Do [Eq- (2111
(d) Sensitivity of fit along Cl critical isotherm to I% perturbation in Eo Fq. (2111
(e) Sensitivity of fit dong C critical isotherm to 1 % perturbation in Fo [Eq.
( f ) Sensitivity of fit along Cl critical isothenn to 1% perturbation in Go CEq- (2111
Figure
3-1 2(a) Temperature dependence of parameter Bo in Eq. (20) modified to include v7 term
Temperature dependence of parameter Co in Eq. (20) modified to include V? tenn
Temperature dependence of parameter Do in Eq. (20) modified to include V? term
Comparison of temperature functions for fitting Bo(T) for methane
Comparison of temperature functions for fitting Co(T) for methane
Comparison of temperature Functions for fitting Do(T) for methane
Spurious inflections in sub-critical isotherms of n-pentane with Eq. (21)
Spurious roots along 433 K isotherm of n-pentane with Eq. (21)
Comparison of errors in pressure along C[ critical isotherm for new EOS and modified B WR equations
Comparison of emors in specific volume dong C1 critical isotherm for new EOS and modified BWR equations
Comparison of errors in pressure along C3 critical isotherm for new EOS and modified B WR equations
4-2 (a)
Comparison of errors in specific volume dong C3 critical isotherm for new EOS and modified B WR equations
Comparison of errors in pressure along a-Cs criticai isotherm for new EOS and modified B WR equations
Comparison of errors in specific volume dong n-Cs critical isotherm for new EOS and modified BWR equations
Comparison of errors m pressure along H2 critical isotherm for new EOS and modified BWR equations
Comparison of errors in specific volume along Hz critical isotherm for new EOS and modified BWR equations
4-5 (a) Comparison of erron in pressure dong C02 critical isotherm for new EOS and modified BWR equations
@) Comparison of errors in specific volume along C02 critical isotherm for new EOS and modified B WR
4-6 (a) Cornparkon of errors in pressure along SO2 critical isotherm for new EOS and modified BWR equations
@) Comparison of errors in specific volume along SO2 critical isotherm for new EOS and modified B WR equations
4-7 (a) Comparison of errors in pressure along H20 critical isotherm for new EOS and modified BWR equations
(b) Comparison of errors in specific volume along H20 critical isotherm for new EOS and modified B WR equations
3-8 Error in saturation pressure for methane
4-9 Error in saturation pressure for n-pentane
4-10 Error in saturation pressure for SO2
4-1 1 Error in Liquid specific volume for methane
4-12 Error in saturated Liquid specific volume for n-pentane
4-13 Error in saturated liquid specific volume for SO?
4-14 Error in saturated vapour specific volume for methane
4-1 5 Error in saturated vapou specific volume for n-pentane
4-16 Error in saturated vapou specific volume for SOz
4-17 Error in vaporization enthdpy of methane
4-1 8 Comparison of EOS exponential terms dong methane critical isotherm
5- 1 Expanded view of the fit of Eq. (29) to calculated values of C o o near the critical temperature for methane
LIST OF SYMBOLS
A at, a29 a3, &b,C B, C, D, E, F, G, H Bo, Co, Do, Eo, Fo, Go, Ho b bo, bl, b?, * * -
el, c2, c3 dl, d2, d3 el, ez, e3 f H KR kl, k?, k3 n P R S T v x7 Y Y z
Greek Symbols a P, Y Y 8 9 E, 4) P? Y, 8 N v 0 P 0
Subscripts/Superscripts c calc EOS ERR =xp lis R REP
EOS constant EOS constants EOS constants Virial coefficients; EOS constants EOS Constants Covolume parameter EOS constants; EOS temperature hct ion constants EOS temperature h c t i o n constants EOS temperature function constants EOS temperature function constants Fugacity Enthdpy Reduced EOS constant Adjustable constants in Eqs. (24), (3 1) Exponent in Eq. (34) Pressure Universal gas constant Regression objective function Temperature Specific volume adjustable constants in Eq. (24) Hard-sphere reduced density Compressibility factor
EOS constant Reduced EOS constants in Eqs. (4), (5) Parameters in new EOS enthdpy expression Eqs. (38), (39) Vaporization enthalpy Fugacity coefficient Molar density Acentric factor Reduced density
Critical Calculated vdue Equation of state Error Experimental Liqyid Reduced Repulsive
res sat
VaP
Residual Saturation Vapour
1.0 INTRODUCTION
Equations of state are one of the most p o w e m tools available for calculating fluid
thermodynamic properties. In its simplest form, an equation of state (EOS) is a
functional relationship between a fluid's pressure, temperature, and density. However,
by applying fundamental thermodynamic relations to the basic EOS form, the departures
from ideality of all thermodynamic properties can be derived as functions of temperature
and density. Equations of state can be applied to both vapour and liquid phases, and
when written in pressure-explicit form, a single EOS can represent coexisting vapour and
liquid phases simultaneously through multiple density roots.
In order to predict fluid thermodynamic and phase-equilibrium properties accurately, an
equation of state (EOS) must first be able to properly represent the pressure-volume-
temperature (PVT) behavior of pure substances. Pure-substance PVT behavior in the
vicinity of the critical point is particularly difficult to reproduce, and an EOS's ability to
fit data in this region depends largely upon its bctional form.
Traditionally, cubic equations of state based on the van der Waals equation have been
most popular for chemical process design work. However, these equations have
limitations that justify consideration of more complex forms. In particular, they are
unable to properIy represent thermodynamic behavior near the pure-component critical
point, especially along the critical isotherm.
Two-parameter cubic equations Like the Peng-Robinson (PR) or Soave-Redlich-Kwong
(SRK) EOS have too few parameters to simdtaneously match all the required conditions
at the critical point. Consequently, they normally match van der Wads' derivative
2
conditions (~P/~v=#P/~~v~=o) by forcing the parameters a and b to match critical
pressure and temperature, at the expense of the predicted critical volume. Saturated
liquid density predictions are poor, and compressed liquid densities are often
unsatisfactory as well.
By contrast, non-cubic equations of state have higher-order density terms or exponential
terms that potentially provide greater flexibility for expressing near-critical behavior.
Non-cubic equations also have many more adjustable parameters than cubic equations.
allowing them to match critical point PVT and derivative conditions, as well as saturation
properties simultaneously. For these reasons, carefully developed non-cubic EOS have
the potential to make better predictions of saturated liquid densities, compressed liquid
densities, and fluid thermodynamic properties.
Historically, non-cubic EOS have developed alongside cubics for over 100 years. During
that time, their greatest drawback has perhaps been that they typically have many more
adjustable constants than cubic EOS, requiring extensive use of data regression to
determine best-fit values for the constants. As a rule, the more constants in the equation,
the more difficult it becomes to not only determine values for pure-component
parame ten, but to develop generalized correlations, assign temperature dependence
fimctions, and establish mixing rules. And aside fkom quartic polynomial equations.
analytical solutions are not available for determining density fiom non-cubic EOS at a
given pressure and temperature. thus robust numerical root-solving procedures are
required. In the past, these difficulties meant that more computer time was needed to
3
develop and use non-cubic EOS than was required for simpler equations, and their use
has consequently been Limited.
Although development, generalization, and implementation of non-cubic EOS all pose
serious challenges, technological advances over the last decade have signScantly
improved the speed and storage capacity of desktop computers, making viable the
commercial application of non-cubic equations of state.
2.0 NON-CUBIC EQUATIONS OF STATE
2.1 The Viriai Equation of State
Although cubic equations of state have traditionally been more popular for chemical
process design applications, non-cubic EOS date back just as far. One of the earliest and
perhaps best-known equations of state is the Virial EOS, first developed by Thiesen as
early as 1885, and modified firher by kun.merlingh-O~es in 1901 (Mason and
Spurling, 1969). While the original basis for the equation was empirical, it was later
shown that the equation could be derived theoretically by considering the effects of
intermolecular forces (Mason and Spurling, 1969).
The Virial EOS is essentially an infinite series in either volume (pressure-explicit form)
or pressure (volume-explicit €om). The pressure-explicit form of the equation is written
RT RTB RTC p=- +-+,+... v v- v
where V is specific volume, and B and C are called virial coefficients. The virial
coefficients take on different values for different gases, but depend only on temperature,
and not on pressure or density (Mason and Spurling, 1969). While the Virial EOS does
have a theoretical basis, it is valid only for the vapour phase at densities below the critical
density (Mason and Spurhg, 1969), and therefore cannot be used to represent vapour-
liquid equilibrium. And although virial coefficients can be determined experimentaIly,
only the second virial coefficient B can be calculated Eom experimental measurements
with reasonable accuracy (Mason and Spurling, 1969).
2.2 BWR-Type Equations of State
Of the many non-cubic EOS developed after the Virial equation of state, the original
Benedict-Webb-Rubin (BWR) EOS and its many variations have perhaps been the best
known and most widely used. The BWR equation can be written as:
where B, C, D, E, and F are parameters (not to be confused with the virid coefficients in
Eq. (1)) whose values, with the exception of F, are temperature dependent. These five
constants are used to express the equation's density dependence at constant temperature.
The temperature dependence of these parameters is as follows:
The BWR EOS was proposed as an improvement to the Beattie-Bridgeman EOS,
enabling it to represent high-density fluid properties more accrtlateIy (Benedict et al,
6
1940). While the Beattie-Bridgeman equation was essentially a quartic polynomiai in
volume with temperature-dependent coefficients, the BWR EOS was unique in that it
introduced an exponential term to better represent liquid-phase hacit ies at high
densities and low temperatures (Benedict et al, 1940). In a sense, the exponential term
accounts for the contributions of higher-order virial coefficients, resulting in a compact
yet flexible non-cubic form capable of accurately fitting pure-substance critical
isotherms, and thus improving PVT predictions-in particular, liquid densities-ver
those of cubic equations.
Many modifications of the B W R EOS have been developed since the original equation
was proposed. One of the best-known was that of Starling (1973), who used the same
volume dependence expressed in Eq. (2), but modified Eqs. (3a-d) to arrive at different
expressions for the temperature dependence of the parameters in Eq. 2. Starling also
introduced a generalization of the pure component parameters in terms of critical
properties and acentric factor (Han, 1972; Starling and Han, 1973).
Other modifications have added terms in powers of density not represented in the original
equation, or have added additional constants in the temperature dependence expressions.
A recent BWR modification was developed by Soave (1995), who proposed the form:
where
Aside from being written in terms of non-dimensional parameters, the essential difference
between Eqs, (2) and (4) is that the fourth term in Eq. (4+6g-is equivalent to having
a fifi-power term in reciprocal volume in the pressure-explicit form of the equation,
instead of the sixth power shown in Eq. (2).
Soave proposed new generalized functions to express the temperature dependence of the
parameters p, y, 6, and E as follows:
8
where p,, y,, and E~ are values calculated at the substance critical temperature, and bl,
b2, cl, cl, c3, el, ez, and e-j can be expressed as functions of acentric factor. Parameter 6 in
Eq. (4) takes on a constant value that is the same for all substances.
[n developing this B WR modification, Soave paid particular attention to fining PVT data
along critical isotherms of the substances of interest, to promote accuracy in the near-
critical region.
Other BWR modifications have included the equation of Nishiumi (1 975), a fifteen-
constant equation that extended the range of applicability of the BWRS equation to lower
reduced temperatures. This equation was later extended to polar pure substances using
three new polar parameters (Nishiumi, 1980).
Platzer and Mauer (1989) also developed a generalization for the Bender EOS (Bender,
L971), a 20-constant modification of the BWR equation. The generalized equation was
applied to polar substances as well, through a polar factor. Their generalization generally
performed better than Pitzer-type interpolation procedures, such as the Lee-Kesler
method as applied to the Bender equation.
2.3 Repulsive Terms
Considerable work has been done recently to better represent the contributions of
repulsive intermolecular forces in EOS predictions. It has long been recognized that the
repulsive term in the van der Wads-type cubic EOSs
does not properly represent the high-density behavior of hard-sphere fluids (Carnahm
and Starling, 1972). One approach to deveIoping a better repulsive term has been
through computer simulation, using simplified intermolecular potential functions to
model molecular interactions. The results are then fit using a semi-empirical expression.
Camahan and Starling (1969) developed an empirical equation that accurately fit results
of computer simulations of molecules that interact via a hard-sphere potentid function.
The equation is written as
where
and b is a covoIume parameter related to the volume occupied by the molecules.
Carnahan and Starling (1972) showed that a complete equation of state can be
constructed by adding a suitable term to the hard-sphere expression to represent attractive
forces. They found that when Eq. (7) was used instead of Eq. (6) in the van der Wads
and Reach-Kwong equations, calculated enthdpy departures and densities were
generally improved for fight hydrocarbons and a few non-hydrocarbons commonly found
20
in natural gas. They also noted some improvement in hgacities for five light
hydrocarbons.
Scott ( 197 1) proposed a simpler hard-sphere repulsive EOS :
which approximates Eq. (7), but facilitates the possibility of a cubic EOS.
2.4 Recent Developments
Most BWR-type equations of state are empirical in nature-emphasis is placed on fitting
experimental data, rather than on attaching theoretical significance to the various terms.
Recently, however, a trend has emerged where new equations o f state are developed
based on theoretical considerations. Generally, the dependent quantity-usually pressure
or Helmholtz energy-is assumed to comprise a series of terms, each accounting for a
particular thermodynamic phenomenon or behavior. Often, the equations include a
repulsive reference term that is based on the results of molecular simulations using some
type of simplified potential function-hard sphere, hard chains, or Lennard-Jones, for
example. Other terms then account for phenomena such as polarity and association.
Once terms have been included for dl desired phenomena, the form of each term is
established. The SAFT EOS (Chapman et al., 1989) is an example of this type of
equation.
I I
Recently, there has been a resurgence of interest in fluids obeying the Lennard-Jones
intermolecular potential hction. Johnson et al. (1 993) carried out molecular-dynamics
computer simulations on Le~ard-Jones molecules, and used a 32-constant modified
BWR EOS to fit the simulated results. Kolafa aad Nezbeda (1994) regressed an equation
combining a hard-sphere repulsive term (slighrly different in form to Eq. (7)) with a
complex multi-constant attractive term.
3.0 DEVELOPMENT OF THE NEW EQUATION OF STATE
A new non-cubic equation of state has been developed to correlate PVT data for pure
substances. Emphasis has been placed on accurate reproduction of PVT data in the
vicinity of pure component critical points.
3.1 Objectives and General Approach
The main objective of the current project was to effort to develop a new non-cubic
equation of state suitable for use in engineering applications, having the following
characteristics :
1 Accurate reproduction of PVT data in the vicinity of pure-component critical
points, more accurately than other commonly used non-cubic EOS of similar
complexity. In particular, superior reproduction of pure-component critical
isotherms using as few adjustable constants as possible.
. * 11. Accurate reproduction of pure component saturation PVT data, using as few
adjustable constants as possible to represent the temperature dependence of the
critical-isotherm adjustable constants described in (i) above.
A firrther objective was to assess how modifying the basic equation to include a hard-
sphere repulsive tenn, either of the van der Waals type [Eq. (@] or the perturbed hard
sphere type pqs. Q, (8), (9)], wouid affiect the accuracy with which the equation could
reproduce pure-component critical isotherm data
13
There are many ways to develop a new equation of state. The approach adopted for the
new equation was to treat the influences of volume and temperature on predicted
pressures separately. The first part of this work focused on establishing a functional form
to express the pressure as a function of volume (or density) at constant temperature.
Once the density-dependent form of the EOS had been established, the effect of
temperature was incorporated by examining how the EOS constants in that form would
have to vary in order to match pure-substance saturation- and supercritical PVT data.
The overall procedure was recursive, because results fkom the temperature dependence
study uncovered weaknesses in the original density-dependent form that had to be
rectified before temperature dependence could be finalized.
3.2 Density Dependence: Fitting the Critical Isotherm
For a pure substance, the critical isotherm is probably the most difficult to fit using an
equation of state. The critical isotherm for methane is shown on logarithmic coordinates
in Figure 3-1. The figure clearly shows the three distinct regions that a successfil EOS
must reproduce: a liquid-like (high density) region to the left of the critical point where
small changes in density result in large changes in pressure, a near-critical region, and a
vapour-like (low density) region to the right of the critical point, where the slope is much
more gentle. Perhaps the most severe restriction imposed on an EOS when fitting the
critical isotherm arises fkom what are commonly called van der Wuufs conditions: the
first and second derivatives of pressure with respect to volume at constant temperature
both take on zero vaiues at the critical point. This can be expressed mathematically as:
Eqs. (10a,b) indicate that the critical isotherm has an inflection point, located at the
critical point. As a result of these two conditions, the critical isotherm is extremely "flat"
in the vicinity of the critical point, and a very flexible equation of state is required to fit
the isotherm within acceptable accuracy while strictly observing the conditions expressed
in Eqs. (10a,b). The EOS must also match ideal gas behavior at low densities. The ideal
gas limit may be expressed mathematically as
The accuracy with which the EOS is able to predict the second virial coefficient serves as
a measure of how well the equation is able to fit the vapour-like region of the PVT space,
not only for the critical isotherm, but over a wide range of temperature.
la theory, the EOS's required flexibility can be attained by expressing pressure as a
polynomial fitnction of volume. In practice, however, attempts to fit critical isotherms in
this way require a large number of polynomial terms which in turn result in a large
number of adjustable coustants. In order to represent saturation PVT data accurately, it is
necessary to allow the values of some of these constants to vary with temperature,
introducing still more constants. As well, the greater the number of polynomial terms
15
that are included in the form used to fit the critical isotherm, the higher the degree of the
polynomial, potentially resulting in undesirable and unpredictable behavior when the
equation is solved implicitly for volume, as well as possible cross-correlation problems.
The net result is that a polynomial equation of state would quickly become unwieldy and
impractical for engineering use. Developing an EOS with only low-degree polynomial
terms and with as few adjustable constants as possible therefore requires consideration of
other types of mathematical functions, such as exponential or rational hctions.
1 10 I00 Reduced Volume
Figure 3-1 Critical isotherm of methane
At the outset of this work, the approach to fitting the critical isotherm had three
components:
Identi@ c o ~ t s that the EOS must observe.
Fit a reference function to the critical isotherm for a particular test substance,
observing the identified constraints.
Develop a function to fit the error between the critical isotherm PVT data and the
pressure predicted by the fitted reference function. The resultant EOS form
would consist of the sum of the reference hc t ion and the error function.
In addition to the van der Wads conditions expressed in Eqs. (lo), and the ideal gas limit
of Eq. (1 I), two additional constraints were identified:
Eq. (12) is essentially a third van der Waals condition, which is suggested by
experimental data (Martin and Hou, 1955). It was thought that inclusion of this condition
would improve EOS predictions near the critical point, helping the EOS to reproduce the
"flatness" of the critical isotherm in this region. It was also known that additional
specifications or constraints will reduce the number of adjustable constants whose values
must be determined fiom empirical data regression. From this perspective, it is desirable
to identify as many different physical constraints as possible. One additional
specification-the second virid coefficient at the critical temperature-was identified as
a possible constraint, but was not investigated untiI alfer the EOS form had been
developed. This aspect of the work is discussed later.
17
Eq. (13) states that the pressure predicted by the EOS must equal the experimental value
of critical pressure exactly, when experimentai values of critical temperature and volume
are used as inputs.
Eqs. (10)-(13) pose five constraints that the new EOS must observe when reproducing the
critical isotherm of a given substance. To meet these constraints, the EOS would have to
have at least five parameters (i.e. four adjustable constants plus the ideal gas term). Even
then, the values of these parameters would be completely determined by the imposed
constraints, leaving no adjustable parameters for data fitting. Regressing the equation to
critical isotherm data would therefore require more than four adjustable parameters.
Remession Procedure
In order to determine substance-specific vdues for the adjustable constants in various
equations of state, and to re-fit existing equations of state to the same data to allow fair
comparisons, it was necessary to develop an appropriate data-fitting method. While
polynomiaI fimctions are linear with respect to their adjustable constants, an EOS
containing exponential terms (such as in Eqs. (2) and (4) ) or rational fuactions (such as
Eqs. (7) or (9)) becomes non-linear in at least some of the parameters. For this reason, a
non-linear least-squares method was chosen as the basis for data regression. Initially, an
algorithm based on the Gauss-Newton method -per and Smith, 1981) was used, but
was eventudy abandoned in favour of a more reliable method based on the Levenberg-
Marquart algorithm mess et aI., 1986).
18
It was necessary to modify the non-linear least-squares algorithm to enforce the
constraints posed by Eqs. (10)-(13). The ideal-gas limit of Eq. (11) was enforced by
including a term of first degree in inverse specific volume such as the first term in the
right-hand side of Eq. (I), with the constant being numerically equal to RT. Initially, the
van der Wads conditions of Eqs. (10) and (1 2) were enforced by including a penalty
Function in the least-squares objective Function that was to be minimized. The penalty
function was computed by determining the associated derivatives numerically at each
iteration, and multiplying them by a large number; minimization of the objective function
thereby had the effect of forcing the three derivatives to take on very small values. Also,
the condition of matching the critical point expressed in Eq. (13) was enforced by
applying a large weighting factor in the objective function to the pressure error computed
at the critical volume.
Using penalty functions and weighting factors to force compliance with the constraints
introduced a tedious trial-and-error procedure that had to be followed to determine
appropriate values of these factors. In addition, these methods did not force exact
compIiance with the constraints-the critical-point derivatives and the difference
between the predicted and actual critical pressure still retained small non-zero residual
vaIues. It was decided that a more effective method would be to enforce these constmints
algebraically by setting the equations for the residual pressure (i.e. the difference between
the actual critical pressure and that predicted by the EOS) and its derivatives to zero at
the critical point, and solving the resuiting system of equations simultaneously. By using
this method to determine values only for constants m the ref-ce polynomial term, the
19
resulting system of equations is linear, and their solution, carried out repeatedly after a
given number of iterations in the non-linear least-squares algorithm, is straightforward.
Although the EOS is written in pressure-explicit form, it is common when calculating
thermodynamic properties or phase equilibrium to specify pressure and solve the EOS
implicitly for the corresponding density. Since the critical isotherm is very flat in the
vicinity of the critical point, small differences in the specified pressure will cause a large
variation in the corresponding density predicted by the EOS. For this reason, accurate
prediction of density in the vicinity of the critical point requires that the EOS's adjustable
constants be determined using a regression procedure that minimizes erron in density as
well as pressure. This procedure is referred to herein as simultaneous regression. One
drawback of the simultaneous regression procedure is that it requires implicit solution of
the EOS lor density at a given pressure. It was found, however, that since the number of
near-critical data points along the critical isotherms of the substances of interest was
small compared to the total number of data points, simultaneous regression had a
relativeIy srndl effect on the h a l values of the adjustable constants. For this reason, the
regression procedure used to develop the new EOS form was based on pressure errors
only. Simultaneous regression was used only once the form of the new EOS was
finalized, to determine final values for the adjustable constants. Constants obtained from
the pressure-regression were used as initial guesses.
For the tint attempt at fitting the critical isotherm, a sixth degree polynomial in volume
was chosen as a reference term
It had been observed in this work that a sixth degree polynomial was able to fit the
critical isotherm of test substances with high accuracy Eom low densities all the way up
to the critical point. Also, the error between the predicted and experimental pressures
appeared very regular at densities greater than critical, indicating that this error could
possibly be fit by adding a relatively simple error term to the reference EOS. By contrast,
when the BWR EOS [Eq. (2)] was fit to the critical isotherm of methane while observing
the constraints of Eqs. (10)-(13), the error was irregular and difficult to fit, making it
difficult to develop a suitable error term. With five adjustable constants plus an ideal gas
term, the sixth degree polynomial was capable of meeting the five constraints of Eqs.
(10)-(13) while still leaving one adjustable constant for regression of the test-substance
PVT data. Methane was chosen as the tea substance, largely because it is the major
component of natural gas, and because it has a reasonable critical compressibility and its
acentric factor is nearly zero.
Smoothed critical isotherm data for methane, obtained fiom an accurate 33-constant EOS
(Angus et al., 1976a), was used for the regression. During the development of the tables
of smoothed properties in that refefence, the authors fit the 33-constant EOS to a large
number of independent sets of experimental data Although the experimental errors for
21
the individual data sets were not reported in detail, the authors did compare the errors
bemeen their equation and the data for several different properties. For single-phase
pressure and density measurements, the majority of the smoothed data was within 1
percent of the experimental values, and much of it was within 0.5 percent. With the
exception of a few points, the smoothed vapour pressures were within approximately 0.4
percent of the data Smoothed saturated Liquid densities were largely within 0.5 percent,
and saturated vapour densities were mainly within I percent of the data except near the
critical point. For enthalpy of vaporization, the maximum difference was 102.8 J/moI at
180 K, which corresponded to 2.54 percent of the experimental value, with the majority
of the points being within 1 percent.
For this work, the values of pressure and volume along the critical isotherm were
interpolated between smoothed values at tabulated temperatures of 190 K and 195 K.
Since the lower isotherm was less than 0.6OC away &om the critical temperature of
190.55 EC, it was expected that errors introduced by the interpolation would be negligible.
Once the reference term had been fitted to the PVT data for methane, the error between
the pressure predicted by the reference term and the pressure from the PVT data was
plotted on a Iogaritbmic scale as a function of volume. At densities greater than the
critical density, the pressure error appeared as a straight h e on the plot, so an error-
fitting term expressing the logarithm of the error as a linear function of volume was
added to the reference term, written equivalentIy as
22
The combination of reference term and error term was then able to fit the critical
isotherm well, except in the immediate vicinity of the critical point.
To fit the error in critical region, a second error term was developed. It was desired that
this term fit the error near the critical point, but not afKect the data fit at sub-mitical
densities, which was already very good using only the reference term. For this purpose, a
fourth degree polynomial in density was used. The density in this term was shifted by an
amount equal to the critical density, in order to take on a value of zero at the critical
point, since exact prediction of the critical pressure-as expressed in Eq. (13)-was a
constraint already imposed on the reference term. In order to force this term to vanish at
densities lower than the critical density where its contribution was not required, it was
multiplied by a "switching function"-essentially an S-shaped curve that switches
between a value of 0 and a value of 1 over a very short range of density in the vicinity of
the critical point.
The resulting overall EOS was obtained by summing the reference term [Eq. (14)], the
k t error term [Eq. (IS)], and the second emr term, and was written as
P = RTp + a2p2 + a3p3 + arp4 + asp5 + asp6
23
Percent error in pressure dong the criticai isotherm of methane is shown for Eq. (16) as a
function of reduced volume in Figure 3-2. The figure shows that while the error is fairly
low at high densities (low reduced volumes), it becomes unacceptably large in the
vicinity of the criticai point, and at low-to-intermediate volumes to the right of the critical
point. It was also decided that the resulting EOS had too many adjustable parameters,
and that inclusion of the second error term (i.e. the density-shifted polynomial multiplied
by the switching function) made the equation too complex. Subsequently, ways in which
Eq. (16) could be simplified were considered.
Closer examination showed that the second term in Eq. (16Fplotted on a logarithmic
scale as a function oEspecific volume-appeared approximately as a straight line. This
indicated that the term could be equivalently represented by an exponential function of
density of the same form as Eq. (IS), but having different values for the adjustable
constants. The modified equation retained the sixth-degree polynomial as the reference
term, and was written
Percent error in pressure along the critical isotherm of methane is shown for Eq. (17) in
Figure 3-3 as a fimction of reduced volume. While the form of Eq. (17) is less complex
than that of Eq. (16), Figure 3-3 shows that its use provides only a slight improvement m
the fit to the data for volumes less than the critical volume, but worsens the fit at volumes
greater than the critical voIume.
-6 -00 0. I 1 10 100 1000
Reduced Volume
Figure 3-2 Error in pressure along C I criticai isotherm for Eq. (1 6 )
At this point, a different approach was taken. First, the degree of the reference
polynomial term was reduced to five by dropping the last term in Eq. (14). As the
reference term now contained only four adjustable constants in addition to the ideal gas
tern, values of these constants would be completely specified by enforcing the conditions
in Eqs. (lo)-( 13). Now, the only adjustable constants available for fitting the data would
be those found in the error terns.
The second step was to re-examine the error t m s in Eq. (17). Before doing so,
however, values for the four adjustable constants in the reference term were determined
for methane by enforcing the conditions, and the pressure error of the reference term
dative to the critical isotherm data was examined, again on logarithmic coordinates. It
was found that for this simplified reference term, the pressure error away hrn the critical
point was a smooth curve-not quite a straight h e , but close enough that a low degree
polynomial in volume, placed inside the exponential, would Likely provide a good fit.
-6.00 0.1 1 10 100 1000
Reduced Volume
Figure 3-3 Error in pressure along Cl critical isotherm for Eq. (1 7)
This effectively allowed the two exponential terms in Eq. (17) to be combined into a
sinde term. Near the critical point, the pressure error curved more sharply, tending to
zero error at the critical point itselt To account for this, extra higher-degree terms were
added to the voiume polynomial inside the new exponential enor Function. Initially, a
fifth-degree polynomial in volume was chosen, leading to another EOS form as fouows:
P = RTp + a,p2 + a,p3 + a,p4 + asp5
26
Eq. (18) was fit to data for methane, and the pressure error-shown in Figure 3-4 as a
function of volume-was again examined. It began to appear that enforcing the third-
derivative condition of Eq. (12) was overly restrictive, and was probably degrading the
quality of the fit in the vicinity of the critical point, rather than improving it. For all
subsequent trials, it was dropped corn use as a specification.
At this point, it was decided as a next step to examine variations on the basic form
expressed in Eq. ( 18), but with fewer adjustable parameters. For both the reference term
and the polynomial term inside the exponential, two approaches were taken: drop the
terms with the highest degree in volume, and drop terms with even powers of volume.
Comparison of these and other variations on Eq. (1 8) led to new forms as follows:
27
Average absolute errors in pressure along the critical isotherm of methane were as
follows: Eq. (19a)-0.78%, Eq. (19b)-0.62%, Eq. (19ck21.8%, Eq. (19d)-0.49%,
Eq. (19+1.13%. The AAD% for Eq. (19e) was lower than that of Eq. (19c), but higher
than the AAD% of the other equations.
1 10 100 1000
Reduced Volume
Figure 3-4 Error in pressure along CI critical isothenn for Eq. ( 1 8)
Of the six equations, Eq. (I9d) resulted in the best fit for methane, and was selected for
fiuther development.
The pressure error exhibited by Eq. (19d) showed considerable improvement over earlier
forms-Eqs. (1 6) and (1 +being essentially the first of the many forms examined that
could fit the critical isotherm of a test substance within acceptable error limits. At this
point, a sensitivity analysis was carried out in an attempt to M e r reduce the number of
adjustable parameters in the equation. Each of the six parameters in Eq. (19d) was
28
perturbed by 2% of its best-fit value, and the effect on the equation's pressure error was
examined It was found that the pressure error was sensitive to perturbations in
parameters a? to ad in the equation, but showed very Little sensitivity to perturbations in
a;--the equation was modified thereafter to exclude parameter a;r, resulting in the
following equation:
Figure 3-5 shows the error in pressure for Eq. (20) along the critical isotherm of methane.
In this figure-and in others presented later in the text for other equations-the e m r
takes on a definite negative bias, especially at reduced volumes greater than unity. This
is mainly because the best-fit value of the second virid coefficient, which is directly
proportional to parameter a*, differs &om the experimental value, as discussed in Section
5.1. Shortly after Eq. (20) was devised, a flaw was identified in its form. The equation
was regressed to critical isotherms for additional substances, and in all cases, it was
observed that the adjustable constants alternated in sign--az took on a negative value, a3
was positive, and a4 was negative. This meant that in the limit, as specific volume
approached a value of zero, the reference polynomial would tend to 4, since the
coefficient of the highest power in specific volume took on a negative value. The
exponential term's zero-volume limit, however, takes on a finite value, equal to the
constant as. The r d t was that the pressure predicted by the overall EOS tended to -oo as
specific volume approached zero.
I 10 100 1000 Reduced Volume
Figure 3-5 Error in pressure along Cl critical isotherm for Eq. (20)
In practical terms, the equation was able to fit the critical isotherm for test substances, but
at some low specific volume-and correspondinply high pressure-the predicted
isotherm would "tum-around", as shown in Figure 3-6. Correcting the problem would
have to rely on one of two solutions:
Modifylng the reference term to take on a zero-volume limit of +a.
Modifylng the error term to take on zero-volume limit of +a, which it would have to
approach more rapidly than the reference t e m approached -a.
Attempts to modify the error term tended to degrade the equation's ability to fit critical
isotherms, and so were abandoned. It was found that the simplest and most effective
solution was to add another volumetric term to the polynomial portion of the equation.
The new term would then have the highest power in reciprocal volume, and as Iong as it
had a positive-valued coefficient, the reference term-and therefore the overall EOS-
would approach the correct Limit as volume tended to zero. After trying different powers
of V, a seventh-degree term was chosen. Fortunately, this added only one extra
parameter to the equation.
0.1 I I 0 100 Reduced Volume
Figure 3-6 Critical isotherm oCC predicted by Eq. (20)
With the new seventh-degree term being added to the equation, the power of the second-
highestdegree term was re-examined. It was found that changing the degree of this term
from f i e to sir reduced the overall average deviation in pressure dong the critical
isotherm of the test substance, methane, and so the change was adopted. At that point,
the basic volumedependent form of the equation could then be written as
This version of the EOS was reported by Kedge and Trebble (1999).
3.2.4 Alternate Re~uisive Terms
For an empirical EOS, greater emphasis is normally placed on the equation's ability to fit
thermodynamic property data, than on the theoretical significance of its terns. This does
not mean, however, that empirical equations cannot represent the contributions of
attractive and repulsive intermolecular forces, at least in an approximate and qualitative
way. Indeed, van der Waals' original cubic EOS was based on the concept that
macroscopic deviations from the ideal gas law could be attributed to the separate
contributions of attractive and repulsive forces. Here, attractive forces were incorporated
into the equation through a term representing a negative deviation from the ideal gas
pressure, while repulsive forces were thought to arise fiom the volume occupied by the
molecules themselves-or erciuded volume-as represented by the "covolurne"
parameter, b. Substituting V-b in an EOS instead of V-and thereby creating a vertical
asymptote at V=b-is perhaps the simplest way of representing the effect of excluded
volume on PVT behavior.
Unlike common cubic EOS, LittIe attention has been paid to representing repulsive forces
directly in non-cubic equations of state. Most B WR-type EOS simply use the ideal gas
equation as the leading term, without incorporating a co-volume parameter. To
determine whether there would be any advantage to including some representation of
repulsive forces m the new EOS-either through a covolume parameter, or a hard-sphere
type term-three alternate versions of Eq. (21) were derived and were fit to the critical
32
isotherm of methane for comparison. The three alternate equations were written as
follows:
where, again, y = b/4V, and b is a covolume parameter. In Eq. (22a,b), repulsive forces
are represented by a van der Waals-type covolume, whereas in Eq. (22c), the repulsive
forces are incorporated by using the Scott Hard-Sphere expression as the leading term in
the equation.
Figure 3-7 shows the error in predicted pressure along the critical isotherm of methane
for Eq. (21), as well as the three alternate versions (Eqs. (22)). Eq. (21) gave the lowest
Average Absolute Deviation (%AAD: 0.379%), mainly because it gave the best fit in the
liquid-like region. All three alternate equations gave better fits, however, in the vapour-
like region (greater-than-critical volumes).
Figure 3-8 shows the error in the predicted specific volume, again for the critical
isotherm of methane. In this case, Eq. (21) gives the highest YoAAD. The pattern is the
same as for pressureEq. (21) performs best m the Iiquid-like region, while all three
alternatives perform better in the vapour-like region. The difference, however, is that
here, the deviations in the vapour-like region are much greater than in the Liquid-Like
region, and make a larger contribution to the %M. For the pressure errors, Iow errors
in the liquid-like region tended to offset the deviations in the vapour-like region.
Comparing the two alternatives where only the leading term was modified to incorporate
hard-sphere repulsions (Eq. (22% c)), both figures show that using the Scott HS term
results in a better critical isotherm fit than simply using the van der Waals covolume.
When a covolume parameter is used, however, a cornparision of the curves for Eq. (22a,
b) shows that better fits are obtained by including the covolume in each term, rather than
in the fmt term only.
I
Reduced Volume
4
__--,*___..---.---.-...*.~---+~~ -New EOS [Eq. (21 )I -*.
I - - - Covolume (first term) [Eq. (22a)j ..*. - - Covolurne (all terms) [Eq. (22b)J -ScottHard-Sphere[Eq.(22c)] --•
t i I
> 1
Figure 3-7 Error in pressure dong CI critical isotherm for alternate versions of the new EOS
1
Reduced Volume
- -New EOS [Eq. (21 )J
- - - Covolume (first term) [Eq. (22a] - - - Covolume (all terms) [Eq- (22b)l
- - Scott Hard-Sphere [Eq. (22c)l ---------- \;/-- -.- -- .- .- .--------- -- .. i
Figure 3-8 Error in specific volume along CI critical isotherm for aiternate versions of the new EOS
It is interesting to note how the regressed values of parameter b vary between the
alternate equations. First, the way the Scott equation [Eq. (9)] is written results in a
vertical asymptote at V = K bSCom. AS a consequence, we would expect the value of
bScorr to be approximately twice the value of that used in the van der Wads co-volume
term, in order for the two terms to have approximately the same asymptote.
Table 3-1 shows the best fit values for parameter 'b' m each of the three alternate
equations, obtained by simultaneous regression along the critical isotherm of methane.
As expected, the value of b in Eq. (22c) is approximately twice that in Eq. (224,
indicating that each of these versions of the EOS would have a vertical asymptote at
approximately the same value of V. The best-fit value of b for Eq. (22b) was lower than
that of Eq. (22a) by approximately a factor of 10. At the same time, this equation yielded
the lowest overall value for the regression objective hction.
Table 3-1 Values of 'b' parameter for alternate versions of the new EOS
-
Equation 'b ' (cm3/mo 11
3 2 . 5 Additional S~ecifications: Second Virid Coefficients and Third Derivatives
Equations (lo), (I 1), and (13) express three constraints that were enforced when fining
candidate equations of state to pure-component critical isotherms. Each consaaint that is
specified reduces by one the number of adjustable constants whose values must be
determined from data regression. Additional constraints are possible, but were not
included in the EOS development phase of the project. In this section, two additional
constraints-enforcing the second virial coeacient at the critical temperature, and
forcing the third derivative of pressure with respect to volume to zero at the critical
point-are investigated to determine their effects on the quality of critical isotherm dts
obtained using the new EOS.
The k t additional constraint considered was the second virial coeEcient, B, which is
the coefficient of the reciprocal-voIume-squared term in the Virial EOS Fq. (I)]. This
term represents the first deviations fiom ideal gas behavior that are observed as pressure
36
is increased. We would therefore expect that forcing the EOS to take on the correct value
of B would improve the equation's fit in the vapour-like region along the critical
isotherm. Conversely, we would expect that if the correct value of B were not forced, but
rather left as an adjustable parameter, accurate fitting would require the value obtained
&om regression to be close to the correct value.
Figure 3-9 shows the deviations in pressure along the critical isotherm of methane, when
Eq. (21) is forced to take on the correct vaiue of B. The relationship between B and the
second coefficient of Eq. (2 1) is as follows:
B, = RTB (224
An experimental value of B was not available for methane at its critical temperature.
Instead, the EOS was forced to match a value calculated &om the correlation of
Tsonopolous (1974). The figures show that forcing the correct value of B greatly
improves the fit along the vapour-like portion of the critical isotherm, but drastically
degrades the fit in the Liquid-like region. This result is not surprising, considering that
Eq. (21) was not specifically developed to fit the error that results when the correct B is
enforced. Another difficulty encountered when B is forced is that the alternating
sequence of signs for successive polynomial terms is not preserved-as a result, the
highest-degree polynomial term in Eq. (21) takes on a negative d u e , which re-
introduces the critical isotherm 'turn-arolmd' problem described in Section 3.2.3. While
these results indicate that forcing a correct value of B is not a successll strategy for Eq.
37
(21), this does not necessarily mean that a suitable error term could not be developed to
replace the existing exponential term.
1
Reduced Vof urne
Figure 3-9 Error in pressure along CI critical isotherm for new EOS with forced second virid coefficient
The second additional constraint examined was to force the third derivative of pressure
with respect to volume to take on a value of zero at the critical point, described by Eq.
(12). As discussed in Section 3.2.1, this constraint was originally included as one of the
conditions to be enforced during development of the new EOS, but was dropped after it
proved too detrimental to fitting pure-component critical isotherms. At this point, the
third-derivative constraint was re-examined to observe its effects on the finished EOS.
Figure 3-10 shows the deviations in pressure dong the critical isotherm of methane,
whe-in addition to the three constraints descn'bed by Eq. (10) and (13)-the third
38
derivative of Eq. (21) is forced to take on a value of zero at the critical point. It is worth
noting that the deviations in both pressure and volume exhibited under these conditions
are similar to those shown in Figure 9, for enforcement of the critical second virid
coefficient. The fit in the vapour-like region to the right of the critical point is very good,
whereas large deviations occur in the liquid-like region left of the critical point.
1
Reduced Volume
Figure 3-10 Error in pressure along CI critical isotherm for new EOS with third derivative forced to zero at critical point
The similarities between the critical isotherm fits obtained with the two constraints
separately occur because forcing the EOS to take on the correct B value results in a third
derivative that is nearly zero. At the same time, forcing the third derivative to zero at the
critical point results in a best-fit B value that is very near the correct value. It is not
known whether this is a general result, applicable to all nou-cubic EOS, or whether it is
unique to Eq- (21).
3 2.6 Sensitivitv Analysis
To understand the behavior of the new EOS along purecomponent critical isotherms, it is
usehl to determine which parameters contribute most to the calculated pressure over
various ranges of specific volume. A parameter sensitivity analysis can help in this
regard.
In the analysis, each EOS parameter is increased one at a time by 1% of its value. The
incremental percent deviation in pressure, or d~flerential enor-that is, the difference
between the pressure error calculated with best-fit parameters and that calculated with the
perturbed parameter-is then plotted as a function of reduced volume. The resulting plot
shows the ranges of specific volume where the contribution of the term in the EOS
associated with the selected parameter is significant to the calculated pressure.
Figures 3-1 1 (a-f) show results of the sensitivity analysis carried out for Eq. (21). Figure
3-1 1 (a) shows that a I% forward perturbation in parameter Bo has its greatest impact in
the liquid-like region near the critical point, at a reduced volume of approximately 0.66.
It also shows that the second-degree term in Eq. (21) makes a significant contribution to
the calculated pressure between VR = 1 and VR = 10, which is as expected, since this tenn
essentially represents the second virial coefficient.
I 10 Reduced Volume
Figure 3-ll(a) Sensitivity of fit along C, critical isotherm to 1% perturbation in Bo [Eq. (21)]
1 10
Reduced Volume
Figure3-ll(b) Sensitivity of fit along CI crit ical isotherm to 1% pemnbation in Ca @5q. (21)]
1 10
Reduced Volume
Figure3-ll(c) Sensitivity of fit dong CI critical isotherm to 1% perturbation in Do [Eq. (2 1 )]
1 t o
Reduced Volume
Fignre 3-ll(d) Sensitivity of 5t dong CI critical isotherm to 1% perturbation in Fo [Eq. (21)]
1 10
Reduced Volume
Figure3-ll(e) Sensitivity of fit along CI critical isotherm to 1% perturbation in Go [Eq. (21)]
1 10
Reduced Volume
Figure 3-ll(f) Sensitivity of fit along CI critical isotherm to 1% perturbation m Eo pq. (21)]
43
Figure 3- 1 1 (b) shows the results of a 1 % forward perturbation in parameter Co. The
EOS term associated with this parameter makes almost no contribution to the calculated
pressure in the vapour-Like region, starting only to contribute significantly immediately
near the critical point itself. This term's contribution to the calculated pressure is greatest
in the same vicinity as the second-degree term, near VR = 0.6 1.
Figure 3-1 1 (c) shows results of the analysis for parameter Do. Here, we see that the EOS
term associated with this parameter again makes virtually no contribution to the
calculated pressure in the vapour-like region. The strength of this tern's contribution
increases with decreasing volume, darting at the critical point.
The differential error at the critical point resulting from a 1 % forward perturbation in Do
is less, however, than born the same perturbation in parameter Co. We can also see that
the peak of the curve has shifted left, indicating that successive polynomial terms make
their maximum contributions to pressure at lower and lower volumes.
Figures 3-1 1 (d) and (e) show results of fonvard perturbations to parameters Fo and Go,
the two parameters associated with the exponential term. Perturbations to either term has
no effect on calculated pressures in the vapour-like region, or at the critical point. The
maximum sensitivity to perturbations in both parameters occurs at approximately the
same value of VRapproxhately 0.44. The figure aIso shows that of these two
parameters, the calculated pressure is more sensitive to perturbations in parameter Go, the
parameter inside the exponential.
44
Finally, Figure 3-1 1 (0 shows results of perturbations in parameter Eo, the coefficient of
the highest-degree polynomial term in Eq (21). The figure shows that this term makes no
contribution to the calculated pressure in the vapour-like region, and very little at the
critical point. The differential error reaches a plateau between VR = 0.39-0.46, and
begins to increase again with decreasing VR, as this EOS term begins to dominate the
pressure calculation.
3.3 Temperature Dependence of EOS Parameters
The first step in the EOS development process, which was the subject of Section 3.2, led
to Eq. (21)-a functional form expressing the variation of pressure with specific volume
at constant temperature. For the equation to be complete, however, it must incorporate
the effects of temperature as well as volume. Eq. (21) was developed by fitting data at
only one temperature-the critical temperature-but, since this isotherm is the most
difficult to fit, we should expect the resulting form to be able to fit data at any other
temperature as well.
Eq. (21) contains six adjustable parameters, whose values can be selected optimally to fit
P-V data along a given isotherm. In developing Eq. (21), best-fit vaIues were determined
for these parameters along the critical isotherm of methane. These values represent the
optimum fit to P-V data at the critical temperature, but only at this temperature. In Eq.
(21), the only term expressing the effect of temperature on pressure is the ideal gas term,
which is cleariy not sufficient to represent temperature effects in red fluids. This is
particularly true at sub-critical temperatures, where calculated isotherms must exhibit
45
multiple volume roots in order to represent both vapour and Liquid phases simultaneously.
By allowing some of the coefficients in the equation of state to vary with temperature, the
EOS can represent both sub- and super-critical isotherms, and can reproduce pure-
substance vapour pressure behavior as well. To complete the development of the new
equation of state, it was necessary to express some of these parameters as analyacal
bct ions of temperature. This was the second stage in the development of the new EOS.
3.3.1 General Approach
The objective of the second part of the EOS development process was to express the
coefficients in the new equation as fimctions of temperature. Before analytical functions
could be developed, it was first necessary to determine how the values of the EOS
parameters would have to vary with temperature. This was accomplished by reproducing
saturation data-vapour pressures, and saturated liquid and vapour specific volumes-for
one test substance at sub-critical temperatures, and by fitting PVT data at supercritical
temperatures. At each temperature where data was available, new values were obtained
for those parameters that were allowed to vary with temperature. Once the variation of
these parameters with temperature had been established, a trial-and-error procedure was
carried out to find anaiytical functions that could fit the parameters as functions of
temperature.
At the beginning of this stage of development, the basic form of the new EOS pq. (21)]
was used. However, as the work proceeded, it became necessary to adopt the m o a e d
form of Eq. (22a), incorporating a van der WaaIs-type covoIume to overcome probIerns
46
with spurious roots at sub-critical temperatures. This decision is discussed M e r in
Section 3.3.5
3.3 -2 Determining Tem~erature De~endence
The first step toward incorporating temperature dependence into the new EOS was to
determine the variation of the EOS parameters with temperature. To determine how the
parameters would have to vary, it was decided to focus on reproducing pure-component
saturation envelopes for sub-critical temperatures, and on fitting single-phase PVT data at
supercri tical temperatures.
At each sub-critical temperature, three conditions must be reproduced-the vapour
pressure, and the saturated vapour and liquid specific volumes. Since the density-
dependent form of the new EOS contains six adjustable pararneters, it was only necessary
for three of these parameters to be made temperature dependent in order to match these
three conditions exactly at each temperature* On this basis, a procedure was devised to
determine the values for three selected parameters at each temperature that would allow
Eq. (21) to match vapour pressure and saturated specific volumes exactly. Values for the
remaining three parameters, which were not made temperature dependent, were held
constant at their critical-isotherm values,
Before proceeding, it had to be decided which three parameters would be temperature
dependent. Since virid coefficients are known to vary with temperature, and some
understanding of how second and third virid coefficients has been established @ymond
& Smith, 1969), the EOS parameters corresponding to these coefficients-Bo and Cr
47
were chosen first. It was decided initially that the third temperature-dependent parameter
should be one of the two associated with the exponential term-either Fo or Go. Since the
exponential term makes a large contribution to the calculated pressure at liquid-like
densities, and since the slope of the linear expression inside the exponential could be
loosely related to the fluid's isothermal bulk modulus-a quantity known to vary with
temperature-Go was selected as the third parameter. It was decided that a second
combination of temperature-dependent parameters-Bo, Co, and &--would be
considered for comparison as well, to ensure the best choice of the third parameter.
To determine values of the three temperature-dependent parameters at each temperature,
the EOS was written twice-once expressing the saturation pressure in terms of the
known saturated liquid specific volume, and once in terms of the saturated vapour
specific volume. M e r rearranging these equations, and substituting one into the other,
we have equations for two EOS parameters in Eq. 22a (Ba and Co) in terms of the
remaining parameters.
where
and x designates either Liquid (L) or vapour (V).
We can then iterate to find the value of a third parameter at which the hgacities at the
saturated liquid and vapour specific volumes are equal. The final result is a set of
parameters Bo, Co, and a third parameter, that exactly matches the experimental vapour
pressure, and saturated vapour and liquid specific volumes.
Following this procedure, the temperature dependence of the two sets of parameters was
calculated for methane. Using these parameter sets, the ability of the EOS to reproduce
subcritical single-phase PVT data was examined. It was found that using Do as the third
term resulted in better reproduction of subcritical isotherms then when Go was used. It
was thereafter decided that Do would be the third temperature-dependent parameter.
When these calculations were carried out originally, the form of the EOS had not yet
been modified to use the sixth power of inverse specific volume instead of the fifth
power, so the temperature calculations were carried out using the fifth-power version,
which is equivalent to Eq. (20) with a v7 term added to it.
At supercritical temperatures (220K. 300K, 4 5 0 5 and 600K), the EOS was fit to single-
phase PVT data by linear least-squares fits, with BoT Co, and Do as adjustable parameters.
Figure 342 (a-c) show the results of the temperature dependence caictdations for Bo, Co,
and Do respectively, carried out with the form of Eq. (20) with a V? term added to it.
The plots are shown on reduced coordinates. In all of the figures, we see that the values
take a sharp upward tum near the critical point, with the effiect being most pronounced for
parameter Co. It is likely that the turn is related to each term's contribution toward
meeting the critical c o ~ t s of Eqs. (10) and (13), and if these c o n ~ t s had not been
enforced, we would expect a snoother variation through the critical region.
This forewarns of a difficulty associated with developing analytical functions to express
the temperature dependence of the three EOS parameters: the unusual behavior near the
critical point would clearly be difficuIt to fit with any reasonable function.
To preserve the critical conditions that were enforced at the critical point when
the critical isotherm was fit-which means exactly reproducing critical values for
the three parameters-we do so at the expense of accuracy near the critical point.
At this point, the challenge was to develop simple functions that could still
accurately reproduce saturation data when forced to match critical values of the
three temperature-dependent parameters.
Reduced Temperature
Figure 3-12(a) Temperature dependence of parameter Bo in Eq. (20) modified to include V' term
Reduced Temperature
Figure 3-t2@) Temperature dependence of parameter Co in Eq. (20) modified to include V' term
-0.5 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Reduced Temperature
Figure 3-12(c) Temperature dependence of parameter Do m Eq. (20) modified to include v7 term
3.3.3 Temperature Function Develo~ment
Section 3.3.2 described the procedure by which the required temperature dependence of
the parameters in Eq. (21) was determined. For each parameter, the procedure resulted in
a set of calculated data expressing the value of the parameter at discrete temperatures. To
complete the equation of state, it was necessary to devetop mathematical functions that
would be capable of fitting the temperature dependence data, referred to herein as
temperature functions.
As mentioned briefly in Section 3.3.2, the behavior of the EOS parameters in the vicinity
of the critical point would be difficult to fit with an analytical function-there wouId
necessarily be a tradeoff between achieving accuracy near the critical point, and
predicting the critical point itself. It was decided that the temperature functions must
reproduce the critical values ofthe parameters exactly, in order to reproduce the critical
point and preserve the enforcement of the van der Wads critical conditions-both of
which had been emphasized during initial EOS development.
The first types of temperature fimctions considered were based on n priori expressions
For second and third virid coefficients. Since parameters Bo and Co essentially
correspond to second and third virial coefficients, it was thought that perhaps known
empirical relations expressing these coefficients as fimctioas of temperature could be
adopted.
One weLt-known empiricaI expression for the second virid coefficient of polar and non-
poIar gases is the corresponding-states correlation of Tsonopolous (1 974). This
52
correlation expresses the reduced second virial coefficient as a polynomial function of
inverse reduced temperature. Tsonopo lo us gives numerical values for the coefficients,
but in this work, the coefficients were taken as adjustable parameters.
A similar empirical corresponding-states correlation for third virial coefficients was
proposed by Orbey and Vera (1983), and was written as a polynomial fhction of inverse
reduced temperature. Again, it was assumed that the expression's coefficients could be
considered adjustable.
Preliminary examination of these two forms revealed that each would contribute five
adjustable constants to the EOS. This number was considered too high, and the idea of
using a prion' virial coefficient forms was abandoned in favor of tinding simpler
expressions.
One characteristic of the two virial coefficient expressions was retained for further study:
the expansion of the virial coefficients in powers of reciprocal temperature, instead of the
temperature directly. An initid screening quickly revealed that the curves expressing
temperature dependence of the EOS parameters [Figs. 3- 12 (a-c)] were fit better using
polynomials in reciprocal temperature. This approach was therefore adopted in the
search for temperature functions.
It was also realized that dividing the temperature-dependent EOS parameter values by
various powers of temperature might be able to smooth the curves shown in Figs. 3-12 (a-
c), making them easier to fit with simple expressions. Based on these considerations, the
first temperature fimctions investigated were of the form:
where KR is the reduced EOS parameter at reduced temperature TL x and y are adjustable
constants, and ki is the ? polynomial coefficient of the temperature function. Candidate
functions were screened by plotting (KR/TRX) VS. TRY, and fitting various truncations of
Eq. (24) to the plots using linear regression. Parameters x and y were varied manually to
identify optimum values.
Initial screening led to selection of x = 0.8 and y = 1.5 for Bo, x = 6 and y = 6.5 for Co,
and x = 0 and y = 1 for Do. For Ba and Do, Eq. (24) was truncated after the fourth term,
whereas for Coy it was truncated after the third term. The leading constant-kl-in Eq.
(24) was expressed as 1-kr-k3-h-. . . to ensure that the critical values of the parameters
were preserved at TR = 1. These considerations led to initial expressions as follows:
54
Initially, the values of the adjustable constants bi, Ci, and di were determined by fitting Eq.
(25)-(27) directly to the h o w n values of Bo, Co, and Do at discrete temperatures, as
calculated by the method described later in Section 3.32.
Although Eq. (25)-(27) gave good fits, it was thought that the non-integer exponents in
Eq. (25) & (26) might not apply well to other substances besides methane. For
comparison, a second set of functions was devised for the three EOS parameters, setting
parameter y = 1 in Eq. (24), and restricting parameter x to integer values only. It was
found that truncating Eq. (24) after the fourth term on the right-hand side was suf£icient
for all three EOS parameters.
After several trials, a second set of expressions was devised, as follows:
The same expression was used for parameters Bo and Do, corresponding to setting x = 1
in Eq. (24). The expression for Co, Eq. (29), is similar to the other two, except that it has
an additional term (the T R ~ term) which results from setting x = 2 in Eq. (14).
Figures 3-1 3 to 3-15 show fits of both sets of expressions, represented by Eq. (25)-(27)
and Eq. (28)-(30), to the calculated values of each of the three EOS parameters as
functions of temperature. These fits were carried out by minimizing the sum of squares
of enon directly, as opposed to the sum of squares of percent errors, because the
program used for these regressions (Microsoft Excel 97) was not able to regress percent
errors. Table 3-2 shows for each temperature-dependent parameter the error between the
known (calculated) reduced parameter value and the value predicted using the fitted
temperature functions. As the table and figures show, the second set of temperature
functions produced a better fit to the temperature-dependent EOS parameters, particularly
at subcritical temperatures. For this reason, and because of their greater simplicity and
lower order, the second set of temperature functions represented by Eq. (28)-(30) was
selected over the origind set.
Table 3-2 Comparison of erron for two sets of temperature functions
--
W t e r Absolute Average hiation Set1 Set 2
[Eqs (uHnll (28)-(30)1 % 0.016 0.0 12 co 0.037 0.023 IXI 0.055 0.026
To complete the comparison, a set of logarithmic expressions was briefly considered,
having the general form
Reduced Temperature
Figure 3-13 Comparison of temperature functions for fitting Bo(T) for methane
Reduced Temperature
Figure 3-14 Comparison of temperame functions for fitting Co(T) for methane
0.4 0.6 0.8 1.0 1.2 1.4 1.6 4.8 Reduced Temperature
Figure 3-15 Comparison of temperature functions for fitting Do(T) for methane
A version ofthis expression was applied to parameter Coy but it was found necessary to
include terms up to and including the fourth power in inverse reduced temperature in
order to fit as well as Eq. (29), and so Eq. (31) was dropped &om fiuther consideration.
As well, a temperature function of the form
was tried for parameter Do. It was found, however, that when this equation-along with
best-fit values for dl, d2, and d3-was used in conjunction with Eq. (28) & (29) to predict
saturation properties of methane, the resulting saturation specific volumes were less
accurate than when Eq. (30) was used for Do. It was then decided that expressions such
as Eq. (32), written in terms of reduced temperature directly rather than inverse reduced
temperaturey would not be considered fkther.
3.3.4 Tem~erature Function Rerrression
While the temperature functions were being developed, values of the temperature-
h c t i o n constants be Ci, and di were determined by fitting the temperature functions to
the calculated EOS parameters at several discrete temperatures. This was necessary in
order to determine the fonns of the temperature functions, but it is not the best way to
obtain values for the temperature-fhction constants. Ultimately, we are more concerned
with matching saturation pressures and specific volumes than with reproducing exact
values of the EOS constants at individual temperatures-calculating the temperature-
dependent vaiues of EOS constants merely represents an intermediate step on the way to
calculating fluid properties. As a result, true best-fit values of temperature-function
constants are obtained only when errors between predicted and experimental saturation
properties are minimized directly.
A regression program was developed to carry out this minimization. The objective
fitnction was defined as
where summations are camed out over all experimental (or smoothed) data points.
Minimization of s2 in Eq. (33) represents a non-hear least-squares regression problem,
for which the Marquart-Levenberg algorithm was again empIoyed. Evaluation of s2
59
entails calculating saturation pressure, saturation specific volumes, and supercritical
pressures at each data point. Other key parameters that must be calculated in the
minimization routine are the derivatives of S' with respect to the regression variables-in
this case, the temperature-function constants. These derivatives were calculated
numerically using a forward difference approximation. Values of the constants obtained
Eom regressing the temperature functions to the EOS parameters calculated at discrete
temperatures were used as initial guesses for the direct regression to the fluid properties
database. This procedure was camed out for methane, and allowed re-regression of the
constants in previously published equations of state, to ensure a fair comparison with the
new EOS.
3.3.5 S~urious Roots
Once the temperature Functions describing the temperature dependence of the new
equation's parameters were developed, the new EOS was fit to two other substances-
sulfur dioxide and n-pentane. While the EOS was being fit to n-pentane, it was found
that the equation produced very large errors in calculated saturation properties for a few
low-pressure points, but performed well otherwise. Further investigation revealed the
cause: the EOS was generating spurious roots at low reduced temperatures. Mead of
exhibiting three volume roots at sub-critical temperatures, the equation had five roots.
In itself, this does not present a problem, as long as the smallest volume root occurs
within the specified tolerance of the experimental saturated liquid specific volume. The
problem lies with the abrupt transition between the three-root and five-root conditions.
60
The additional roots do not just suddenly appear within the calculated satwation
envelope, but rather begin as an inflection in the isotherm at pressures somewhat above
the saturation pressure. The inflection represents a non-physical condition, but since the
magnitude of the inflection is mall at first, it does not actually contribute extra roots at
the saturation pressure of interest. As the temperature is decreased, the magnitude of the
inflection becomes larger and larger until finally, the bottom of the inflection intersects
the saturation pressure, giving rise to two extra roots. Isotherms exhibiting these
inflections are shown in Figure 3-16 (a,b) for Eq. (21). In Figure 3-16 (a), the inflections
do not give rise to spurious volume roots at the vapour pressure until the temperature fails
below approximately 433K. Figure 3-16 (b) provides a magnified view of the 433K
isotherm, and clearly shows the spurious roots in relation to the actual physical roots.
This behavior presents two problems. First, within the range of temperatures between
where the inflection in the isotherm first appears, and where the inflection actually
intersects the saturation pressure, the calculated isotherm does not properly represent the
true thermodynamic behavior of the fluid-multiple volume roots are not actually found
along red fluid isotherms outside the two-phase region at pressures above the saturation
pressure.
The second problem is that if the temperature functions are smooth and continuous across
the temperature at which the sudden transition occurs between the three- and five-root
conditions, the root representing the saturated Liquid specific volume will not correspond
to the lowest voIume root, but rather to the third-lowest root, As such, the new roots are
spurious and not physically possible.
-2000 0 0.1 0.2 0.3 0.4 0.5 0.6
Reduced Volume
Figure 3-16 (a) Spurious inflections in sub-critical isotherms of n-pentane with Eq. (2 1)
1
Reduced Volume
Figure 3-16 @) Spurious roots dong 433 K isotherm of n-pentane with Eq* (21)
62
It is conceivable that a set of values could be chosen for the temperature-dependent EOS
parameters so that the lowest volume root does correspond to the experimental saturated
liquid volume, but this would Likely result in discontinuities in the temperature Functions.
And even if such discontinuities were acceptable, this would not eliminate the problem of
inflections in the isotherm above the saturation pressure.
At this point, it was decided that the new EOS should be modified to prevent spurious
root formation. These roots form because the large negative contributions to pressure
£iom some polynomial terms in the EOS briefly dominate the calculated pressure, until
the specific volume becomes low enough for the v7 term-which has a positive-valued
coefficient-to cancel this negative contribution and again introduce the proper slope.
The first approach to eliminating the spurious roots was therefore to increase the value of
the coefficient of the V' term at the temperatures where the inflections would normally
occur. This would compensate for the increasing negative contributions of the Vd tern
with decreasing temperature, thus avoiding the formation of isotherm inflections and
spurious roots altogether. Since adding new parameters to the EOS was undesirable, it
was decided that the temperature dependence of the term would be represented
simply by dividing the term by the reduced temperature raised to an unspecified power n:
The value of the exponent would then be determined by trial and error as the minhnum
value required to eliminate inflections in the isotherm. This type of ternperatme
63
dependence is similar to that found for the \T' term in the Soave-BWR equation, and for
the exponential term in the BWRS equation. For n-pentane, it was found that a value of
n=2 was sufficient in Eq. (34) to eliminate spurious mots. This created a new problem,
however: for both methane and n-pentane, the average absolute deviations in the
calculated saturation properties were now higher than both the BWRS and Soave-BWR
equations. Since this would eliminate the new EOS as a candidate for M e r
development, other solutions were sought.
The next solution that was investigated was to incorporate a van der Wads-type
covolume into the equation. It was thought that if the EOS had a vertical asymptote at a
volume somewhat greater than zero, the spurious roots would not be as likely to form.
Versions of the new EOS with covolumes and hard-sphere terns were already
investigated and described in Section 3 2.4. It was decided that the version having a
covolurne in only the first term would be considered. Of the three equations examined in
Section 3.2.4, Eq. (22a) did not produce the best fit to the critical isotherm of the test
substance, methane. However, the other equations were ruled out for two reasons:
Eq. (22a) is mathematically simpler than the hard-sphere version [Eq. 22~1.
The best-fit value of the covolume parameter in Eq. (22b)-which incorporates the
covolume into the denominator of each polynomial term-is an order of magnitude
lower than that of Eq. (22a). With the vertical asymptote occurring at such a small
specific volume, it is unlikely that using Eq. (22b) would overcome the spurious root
problem at all.
64
In any case, it was shown in Section 3.2.4 that incorporating a covolume even in the first
term improved the fit of the critical isotherm for methane. And using a covolume instead
of allowing the r7 term to vary with tempera- has the benefit of adding only one extra
parameter to the EOS-one that is temperature independent, physically sensible, and
easily generalizable.
For the investigation, values of the EOS parameters were determined by fitting the EOS
to saturation data and supercritical PVT data by the same procedure used originally for
Eq. (31), descnied in Section 3.3.4. The temperature hct ions were not modified-the
forms remained as written in Eqs. (28)-(30).
Subcritical isotherms of n-pentane, calculated using the temperature-dependent form of
Eq. (22a), were examined. It was found that with the covolume parameter, spurious roots
and other isotherm inflections do not occur within the range of the data used in this work.
At this point, the form of the new EOS was finalized, using Eq. (22a) to express the
equation's volume dependence, and Eqs. (28)-(30) to express its temperature dependence.
It is important to note that although incorporating a covolume into the first term of the
EOS prevented spurious roots from forming in the three substances investigated here, this
solution may not work for all substances. in particular, during regression of the EOS to
pure-substance criticaI isotherms, the best-fit covolumes for some substances-water and
COrtended toward negative values. Since negative covolumes have no physicd
meaning, the regressions were repeated, this time limiting the covolume to positive
65
values. In both cases, the best-fit vahe tended toward zero, and this may expose these
substances to spurious root problems.
It is also worth mentioning that a second unrelated cause of spurious roots was
discovered as well. la the regression procedure €or determining values €or the EOS
parameters, the equation is fit not only to saturation data, but to PVT data at supercritical
temperatures as well. The supercritical PVT data was included to ensure that the EOS
would perform we11 as supercritical temperatures. It was found, however, that when the
regression was carried out without the supercriticd PVT data, the EOS performed poorly
at supercriticd temperatures-so poorly that spurious inflections appeared along the
liquid-like portions of the isotherms. These inflections were severe enough to give rise to
multiple volume roots, which in reality do not exist at supercritical temperatures. When
the supercritical PVT data was included in the regression, no such inflections appeared.
3.4 Thermodynamic Properties from the New EOS
The most important use of an equation of state is to calculate fluid thermodynamic
properties. Expressions for these properties as functions of temperature and specific
volume can be derived using fundamental thermodynamic relations between properties-
hgacity, enthdpy, or entropy, for example-and state variables: pressure, temperature,
and specific volume. The problem with these fundamental relations, however, is that
they are normally expressed in terms of integrals and derivatives of state variables. The
EOS provides an analytical relationship between pressure, temperature, and specific
66
volume, which allows us to derive direct functional expressions for thermodynamic
properties fkom the hdamental relations.
Like any equation of state, the new EOS [Eq. (22a) and Eqs. (28)-(30)] can be used to
derive expressions for thermodynamic properties. And because these expressions are
derived using a unique PVT relationship (the EOS), the expressions for the properties are
unique to that EOS. In the present work, expressions were derived for two
thermodynamic properties: fugacity, required to calculate vapour pressures to determine
the temperature dependence of EOS parameters, and enthalpy, used to assess the
adequacy of the temperature dependence of the new EOS's parameters.
3.4.1 Fueaci
Pure-component fugacity is obtained using the following hdarnental thennodyamic
relation:
from which the pure-component fugacity expression for the new EOS can be written:
where t) is the fugacity coefticient, defined asfP, and Z is the compressibility factor.
3-42 Enthal~v Residual
Enthalpy residual-the deviation of a real fluid's enthalpy fiorn the corresponding ideal
gas value-can be derived &om the following fundamental relation:
leading to the enthalpy departure expression for the new EOS, which, after algebraic
rearrangement, is written as
In these equations, Hm is the enthalpy residual, H'"' is the ideal gas enthalpy at the
temperature of interest, and H is the actual enthalpy of the fluid at the temperature and
pressure of interest.
68
The enthaipy of vaporization at a given saturation temperature can then be calculated as
the difference between the saturated--liquid and saturated-vapour enthdpy residuals at that
temperature:
69
4.0 COMPARISON WITH PREVIOUSLY PUBLISHED NON-CUBIC EOS
To compare the new EOS with previously published equations, several different
thermodynamic quantities were calculated for a number of substances. The new equation
was compared to both Starling's version of the B W R equation (BWRS: Starling, 1973),
and Soave's modification (SBWR: Soave, 1995).
The BWRS equation used here was modified slightly: in Starling's original version, two
of the parameten used to express the temperature dependence of the parameter C in Eq.
(2) are also used to express the temperature dependence of the coefficient D, as follows:
The effect of this constraint is that these two parameters are not independent of one
another. In the Soave BWR equation, however, these parameters are independent. Since
one of the goals of this work was to identify the best functional form for fitting pure-
component critical isotherms and saturation properties, it was decided that the
comparison would be more reliable if parameters C and D were independent of one
another in the BWRS equation, rather than being constrained to vary together m a
seemingly arbitrary way.
70
This move makes the BWRS equation slightly more flexible, and ensures that any
deficiencies in its performance compared to the other two EOS cannot be attn'buted to an
explicit dependence between C and D.
Starling's expressions for the temperature hctions were also rearranged to force
reproduction of the critical values obtained fiom regression of the critical isotherm. This
ensures that the critical point is reproduced exactly, and that the van der Wads conditions
Eq. (10a,b)] are preserved at the criticai point. The temperature functions used for the
modified B WRS equation were as follows:
4.1 Pure-Component Criticrl Isotherms
The new EOS [Eq. (22a). Eqs. (28)-(30)], and the two BWR modifications were fit to the
critical isotherms of seven substancet+methane (Angus et al., 1976a), propane
71
(Goodwin and Haynes, 1982), n-pentane (Canjar and Manning, 1967), hydrogen
(McCarty et al., 198 11, carbon dioxide (Angus et al., 1976b), sulfiu dioxide (Kaag et al.,
1961), and water (Keenan et al., 1978)-in order to compare average and maximum
absolute deviations in both pressure and volume from smoothed data or experimental
data. The three critical conditions, represented by Eq. (10) and (13), were enforced in the
fitting procedure. Figures +1(a,b) through 4-7(a,b) show errors in pressure and specific
volume along the criticai isotherms of the seven substances of interest. Table 4-1 shows
the results of the comparisons in terms of average and maximum deviations.
Comparing Average Absolute Percent Deviations (%Am) in Table 4-1, we see that for
pressure, the new EOS has the lowest values overall, while for specific volume, the
SBWR equation has the lowest overall values. For specific volume, the SBWR equation
is followed very closely by the new EOS. For BWRS, the overall %AADs-averaged
over all seven substances-were higher than those of both SBWR and the new EOS. For
all three equations, average values of %AAD were less than 1 percent in both pressure
and specific volume.
The new EOS had the lowest overall Maximum Absolute Percent Deviation (%MAD) for
pressure, while SBWR had the highest. For specific volume, the new EOS's value was
also lowest, followed by SBWR. For both pressure and specific volume, the new EOS
had lower %MAD values than the BWRS equation. Maximum deviations tended to
occur in the immediate vicinity of the critical point.
1
Reduced Volume
- Soave BW R (I 995) - - - BWRS (Starling 1973)
L
Figure41 (a) Comparison of errors in pressure dong Cr critical isotherm for new EOS and modified B WR equations
1 Reduced Volume
- -New EOS
- Soave BWR (1 995) - - * - BWRS (Starling 1973) i
. b
h
Figure 4-1 (b) Comparison of errors in specific volume dong Ci critical isotherm for new EOS and modified BWR equations
1
Reduced Volume
-New EOS - Soave-BWR (1995) "
- - - BWRS (Starling 1973)
. * .. 11 4
Figure 4-2 (a) Comparison of errors in pressure along C3 critical isotherm for new EOS and modified B WR equations
1 Reduced Volume
-.-....***.---***.~--*****.****.**---,-----.*--~-*-*-**..****-**********--*---*+*-*--*-. . i
. . I
C
., -New EOS ~ ~ ~ . ~ ~ ~ ~ ~ , ~ - - ~ ~ ~ ~ , I I ~ . I I . . I I . ~ . ~ . . . ~ - - ~ - ~ - - . - Soave BWR (1 995) ' .* * . - - - BWRS (Starting 1973) -.... i .......-...-' --------.----- - .-*-.--*-.- - .--.
t
Figure 4-2 @) Comparison of erron in specific voIume dong C3 critical isotherm for new EOS and modified BWR equations
1
Reduced Volume
- Soave-BW R (1 995) -.*.................--..-*-*-*LLL I* .----. . - _ - BWRS (Starling 1973) " I
Figure 4-3 (a) Comparison of errors in pressure along n-C5 critical isotherm for new EOS and modified BWR equations
I
Reduced Volume
.
-...-*----*------*----**-----*--.----.*-.-.->--~*..*-.---.----,~....-.-*-*------**.*--.* . *\
-New EOS 8 ' . C
-. - SoaveBWR (1995) c~.,~,,.,--------A-J-~------*------------*----- b #
- - a BWRS (Starling 1973) i . ;
I
Figure 4-3 @) Comparison of errors in specific voIume along n-Cs criticaI isotherm for new EOS and mowed BWR equations
>
*
1
B - Soave-BW R (1 995) -.---..*-------------r-.---c-.*-r,b-----.;--**-*
b * b : - - - BWRS (Starling 1973) "
I r . :
. *
1
Reduced Volume
Figure44 (a) Comparison of errors in pressure along H2 critical isotherm for new EOS and modified B WR equations
I
Reduced Volume
* - :
,*-.- New EOS - C
- Soave BWR (1 995) 8
---. - - - .
A I
Figure 4-4 (b) Comparison of errors in specific volume dong Hz critical isotherm for new EOS and modified BWR equations
1
Reduced Volume
' : - - - - - - r ~ ~ ~ r r . r + ~ ~ . . . . . . - . . ~ - - ~ . ~ ~ . - ~ .
- Soave-BWR (1 995) - - - BWRS (Starling 1973) -
t
Figure 4-5 (a) Comparison of errors in pressure along COz critical isotherm for new EOS and modified B WR equations
-------*--.*--.-*---------.--.------ .--.-,-*--*-* - Soave BW R (1 995)
r i - - - BWRS (starling 1973) - 4 r
I
Reduced Volume
Figure 4-5 @) Comparison of erron in specific volume dong COz critical isotherm for new EOS and modified BWR equations
1
Reduced Volume
-New EOS - Soave-BWR (I 995)
BWRS (Starling 1973) "
Figure 4-6 (a) Comparison of errors in pressure dong SO2 critical isothenn for new EOS and modified BWR equations
-New EOS - Soave BWR (1 995)
BWRS (Starling 1973)
1
Reduced Volume
Figure46 @) Comparison of errors in specific volume along S@ critical isotherm for new EOS and modEed BWR equations
1
Reduced Volume
-Soave-BWR(1995) " - - - BWRS (Starling 1973)
Figure 4-7 (a) Comparison of erron in pressure along HzO critical isotherm lor new EOS and modified BWR equations
1 Reduced Volume
_. - New EOS
- Soave-BWR (1995) " - , , BWRS (Starling 973)
Figure&?@) Comparison of erron in specific voIume dong HzO critical isotherm for new EOS and modified B W R equations
Table 4 1 Comparison of erron between predicted and experimental pressures and specific volumes dong pure substance critical iso thenns
Substance EOS Pressure Volume %AAD %MAD %&ID %MAD
Me ane New EO 0 0.357 2,775 BWRS 0.44 1 1.306 0.386 3.1 16 Soave-B W R 0.948 3,813 0.3 14 1.359
Propane New EOS 0.345 0.972 0.276 1.326 BWRS 0.427 1373 0.5 13 3.767 Soave-B WR 0.302 0.843 0.248 1.679
n-Pentane New EOS 0.65 1 4.527 0.330 1.088 BWRS 0.786 5.061 0.63 7 3 -3 65 Soave-B W R 0.652 4.858 0.393 1.567
H2 New EOS 0.499 I .475 0.378 2.655 BWRS 1,357 4.11 1 0.836 3.769 Soave-BWR 0.708 2.022 0.5 1 6 2,653
C02 New EOS 0.870 6.830 0.442 4.775 BWRS 0.665 7.093 0.498 6.028 Soave-B WR 0,582 7.574 0.376 6.417
SO2 New EOS 1.053 7.525 0.344 1.800 BWRS 1.135 7.5 15 0.576 2.5 1 1 Soave-B W R 1.057 7.773 0.40 8 1.898
Water New EOS 0.71 1 2.400 0.630 7.0 14 BWRS 0.409 2.933 0.519 7.127 Soave-BWEl 0.415 2.923 0.519 7.0 18
Average New EOS 0.645 3.603 0.408 3.062 BWRS 0.746 4.242 0.566 4.24 2 Soave-BWR 0.666 4.258 0.396 3 227
For dl three equations, the largest maximum deviations (%MAD) in pressure occurred
for Cot and SOz, while the largest %MAD in specific volume occurred for water.
4.2 Saturation Properties
Using the new EOS and the two BWR modifications, saturation properties were
calculated for three substances: methane (Angus et d., I976a), n-pentane (Canjar and
M*gJ 1967), and s u b dioxide (Kaag et d., 196 I), chosen respectively to represent
light hydrocarbons, intermediate-weight hydrocarbons, and polar substances. Selected
saturation properties were saturation pressure, saturated vapour specific volume, and
saturated Liquid specific volume. The adjustable parameters of each EOS were
determined using the least-squares procedure described in Section 3.3.4. Table 4-2
shows average and maximum deviations tiom data for the three equations.
Table 4-2 Comparison of erron between predicted and smoothed experimental saturation properties of methane, n-pentane, and s u l h dioxide
Substance EOS Saturation Specific Volume Pressure Saturated Liquid Saturated Vapour
AAD% MAD% W% MAD% AAD% MAD% Methane New E O S ( E ~ . 22a) 0.1 3 0.27 0.17 2.67 0.32 4.03
BWRS 0.17 0.38 0.34 3.76 0.39 5.20 Soave-B WR 0.02 0.08 0.36 3.31 0.24 322
n-Pentane New EOS (Eq. 22a) 1. I4 3 -09 1.08 2.66 1.70 10.83 BWRS 3 3.70 0.70 1.47 1.79 6.63 Soave-B WR 0.33 0.56 0.19 0.81 0.56 4.90
S u l k New EOS (Eq. 22a) 0.27 0.68 0.89 2.29 1.89 8-14 Dioxide BWRS 4.57 8.47 1.97 3.50 6.36 11.85
Soave-BWR 0.43 1.33 0.3 1 1-02 0.62 1.23
For all three substances, the Soave BWR equation gave the best overall fits. In terms of
average deviations, the new EOS generally performed better than the BWRS equation,
with the exception of the saturation pressure and saturated liquid volume for n-Pentane.
For maximum deviations, the new EOS petformed better than BWRS as well, except for
the saturated liquid and vapour specific volumes for n-Pentane. The only two cases
where the new EOS performed better than the Soave BWR equation were for the
81
saturated Liquid volume of methane and the saturation pressure of sulfur dioxide-both
average and maximum deviations.
Figures 4-8 to 4-16 show percent error as functions of temperature for saturation
pressure, and saturated liquid and vapour specific volumes, for the three substances of
interest.
Generally, the deviations in saturation pressure for all equations and substances (Figures
4-8 to 4-10) tend to oscillate smoothly between positive (overprediction) and negative
(underprediction) errors. the difference between equations being the magnitude of the
oscillation peaks.
-New EOS
- - - BWRS (Starting 1973) : I t t f
'U. JU
0.45 0.55 0.65 0.75 0.85 0.95 1.05 Reduced Temperature
Figure 4-8 Error in saturation pressure for methane
-5.0 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05
Reduced Temperature
: -New EOS --. - Soave BW R (1 995)
*--
- - - BWRS (Starling 1973)
I I t t I
Figure 4-9 Error in saturation pressure for n-pentane
-8 -0 0.70 0.75 0.80 0.85 0.90 0.95 1.00 f -05
Reduced Temperature
* - - t -.*-.bt~-..-.:.-.-~--*--.-.---. -----------6----------*a :
4 . *!. & .
k
r . w-*-*-*-*-*-,*-..*---***-r----**-*---. - * - . * * - * - - * * * * * - m . * .
: - Soave BWR (t 995) ; : *
-*--****-*--- - - WVRs (Stadine 1973) ---4 -**----.---- r C -New EOS + C
: * -**-*-*.*--*,-.----*----*+-----*---*;*~*~~*~*.~~.;-~~~~---*-**>**-**-*--*-.>--~---*----*
f . t
. I t t t r
Figure 4-10 Error in saturation pressure for SO2
: ' -New EOS :
--------- . - - - BWRS (Starling 1973)
L .
1 t I -2.0 0.45 0.55 0 -65 0.75 0.85 0.95 1,05
Reduced Temperature
Figure 4-1 1 Error in liquid specific volume for methane
3.0
2.0
1.0 C I
g 0.0 W
-1 .o
-2.0
-3 .O 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05
Reduced Temperature
-New EOS --- - BWR (1 995)
- - - BWRS (Starling 19
, I 1 I 4
Figure 4-12 Error in saturated Liquid specific voiume for n-pentane
-4.0 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00 1.05
Reduced Temperature
.
- Soave BW R (1 995)
-New EOS 1 I 1 I I I I
Figure 4-13 Error in saturated liquid specific volume for SOt
-6 .O 0.45 0.55 0.65 0.75 0.85 0.95 1 -05
Reduced Temperature
-New EOS --.*.-
-**--* - - - BWRS (Starling 1973) ;
Figure 4-14 Error in saturated vapour specific volume for methane
L
-New EOS - - - Soave BWR (1995) ; -- - - - BWRs (Starling 1973)
t 1 f I I I I
Reduced Temperature
Figure 4-15 Error in saturated vapour specific volume for n-pentane
-20.0 070 0.75 0.80 0.85 0.90 0.95 1.00 1-05
Reduced Temperature
s ,,,-, *,**..*i.--* .--- ,----:-* -*------ *-; -----. **--*-:*--*---*--*'!*.*--** --*- i ---------*-- . ' 8
8 ' : * . : * : - . : / : ' : . --------~-,-~-.----------~--4-------- .~------------~------------ i • , . -* . - 4 - - ;
- Soave BWR (1995) . - - - BWRS (Starling 1973) "
. -New EOS . I
. I I
Figure 4-16 Error in saturated vapour specific voIume for S&
Deviations for saturation specific volumes are generally not as regular. As shown in
Figures 4-1 1 to 4-15, the error c w e s for both saturated Liquid and vapour specific
volumes exhibit sharp peaks at subcritical temperatures immediately below the critical
point. For most equations and substances, the maximum deviations over the range of
reduced temperatures considered occurred at these peaks, with the exception of the
BWRS equation for SO2. At subcritical temperatures away from the near-critical peaks,
the deviations tended to oscillate between positive and negative values.
4.3 Single-Phase PVT
Pressures along supercritical isotherms were calculated for three substances-methane,
sulfur dioxide, and n-pentane-using the new EOS and the two BWR modifications, and
were compared to smoothed and experimental data for those substances. Table 4-3
shows the results of the comparisons.
Table 4-3 Comparison of errors between predicted and experimental pressures along supercritical isotherms of methane, pentane, and sulfur dioxide
Substance Max Tr Max Pr New EOS (Eq. 22a) BWRS Soave-B WR AAD% MAD% AAD% MAD% AAD% MAD%
Methane 2.36 152 4.26 26. f t 1.29 5.28 0.57 5.13 n-Pentane 1.73 7-15 2.00 14.98 1.12 8.81 4.3 1 15.60 S d f h Dioxide 1.21 4.05 0.56 3.2 1 4-10 28.43 4.96 18.37
The table shows that in terms of both average and maximum absolute deviations, the new
EOS performs better than the Soave BWR equation for two of the three substances
(sulfur dioxide and n-pentane), but perform better than the BWRS equation for only one
of the three substances ( s u k dioxide). The maximum errors tended to occur on the
steep liquid-like portions of the isotherms, and at high pressures.
4.4 Enthalpy of Vaporization
The enthalpy of vaporization was calculated For methane using the new equation as well
as the two modified BWR equations. Table 4-4 shows the average and maximum
absolute errors between calculated values and smoothed data for the three EOS. While
the Soave BWR equation had the lowest average deviation, the AAD for new EOS was
very close behind. The AAD for the BWRS equation was somewhat higher than for the
other two equations, but the AAD for all three equations was small compared with the
average magnitude of the vaporization enthalpy. The new EOS had the lowest maximum
deviation, with the Soave BWR equation close behind. The MAD lor the BWRS
equation was considerably higher than for the other two equations.
Table 4-4 Error in vaporization enthalpy for methane - -
EOS Vaporization Enthalpy AAD (J/rnoI) MAD (J/mol
New 20.0 176.5 BWRS 24.8 242.6 Soave BWR 15.2 182.8
Figure 4-1 7 shows the error between calculated vaporization enthalpy and smoothed data
as a function of reduced temperature for methane. For all three EOS, the error oscillates
between positive and negative values for the I11 range of temperatures, except near the
critical point. The maximum errors o c c d in the immediate vicinity of the critical
point, where al l three EOS tended to underpredict the vaporization enthdpy.
'- New EOS
Reduced Temperature
Figure 4-17 Error in vaporization enthalpy of methane
4.6 Comparing Exponential Terms
Upon £ k t inspection, the new EOS may appear similar to the BWR equatiowboth are
essentially polynomial expansions in specific volume added to an exponential term. This
similarity is superficial, however, because the exponential term in the new equation
serves a different purpose than the one in the BWR equation and its modifications.
Figure 4-18 shows the contributions to calculated pressure of the exponential terms for
three equations-the new EOS [Eq. (22a)], the BWRS equation, and the Soave BWR
equation-as functions of reduced volume dong the critical isotherm of methane. The
critical isotherm is superimposed on the figure to show the relative magnitude of each
exponential term compared to the actual pressure that the EOS must predict
1
Reduced Volume
Figure 4-18 Comparison of EOS exponential terms along methane critical isotherm
The figure shows that in the two B\KR modifications, the exponential term makes its
most significant contribution near the critical point, and in the liquid-like part of the near-
critical region. For the BWRS equation, the exponential term contributes 95.3 percent of
the calculated critical pressure, meaning that the remaining terms make only a minor
positive contribution equal to 4.7 percent of the computed value. For the Soave-BWR
equation, the exponential term makes an even larger contri%ution at the critical point,
equal to 154.2 percent of the calculated pressure. This means that the sum of the
remaining terms in the equation make a negative contribution equal to -54.2 percent of
the calcuIated value.
As specific volume is decreased below the critical value, the exponential terms in these
two equations rise to reach a maximum, and then fall off again as the critical isothenn
90
becomes steeper. This maximum occurs at a reduced volume of approximately 0.58 for
methane.
By contrast, the exponential term in the new EOS makes almost no contniution at
specific volumes equal to or greater than the critical volume, and does not start making a
significant contribution to the pressure calculated by the EOS until the reduced volume
falls below approximately 0.75. Below this value, the exponential term increases rapidly,
reaching a value equal to 100 percent of the calculated pressure at a reduced volume of
0.47. In this way, the purpose of the exponential term is to improve the equation's fit
along the steep liquid-like branch of the isotherm. This is quite unlike the role played by
the exponential term in the modified BWR equations, where it contributes IittIe to the
calculated pressure along this portion of the isotherm.
5.0 DISCUSSION
For a new EOS to be useful, it must have some redeeming feature that makes it preferable
to use over other established equations. No comparison of equations of state would be
complete without considering how they perform for fluid mixtures-by far the most
common EOS application. But if a new EOS does not clearly show some improvement
over established equations for predicting the properties of pure components, then there is
no point developing it fbrther to apply to mixtures. It would instead be better to focus on
improving the generalizations or the mixing rules for existing equations. For this reason,
the first stage in developing a new empirical EOS should center exclusively on fitting
pure component properties. This was the focus of the present work.
Developing a new EOS that performs better than existing equations is more difficult than
it may seem. The difficulty is that while the EOS itself is written in terms of pressure,
volume, and temperature, it must also make accurate predictions of thermodynamic
properties when fundamental thermodynamic relations are applied to it. And considering
the number of different properties that the EOS must be able to predict over a wide range
of pressure and temperature-including the region near the critical point-it is not hard
to understand why this is difficult even for a non-cubic EOS with I0 or more parameters.
But the greater the number of parameters in the EOS, the more difficult it will be to
develop generalized correlations and mixing d e s for the equation. The result is that a
new non-cubic EOS must strike a balance between accuracy-wbich generally tends to
increase the number of parameters-and ease of application to mixtures, for which fewer
parameters is better.
92
In the present work, an attempt has been made to identify the non-cubic EOS € o m that
best strikes this balance, with the intention of selecting one form-ither the new EOS or
an existing BWR modification-for further development for application to fluid
mixtures. And while the new EOS clearly showed improvements in some areas over two
modified BWR equations that were selected for comparison, the results of the
comparisons did not point unambiguously toward any one of the three equations. A
discussion of the relative merits and drawbacks of the new EOS follows herein, in terms
of the major stages of its development.
5.1 Fitting the Critical Isotherm
Establishing density dependence by fitting the critical isotherm of a pure component is
perhaps the greatest challenge in developing a non-cubic equation of state. And by far
the most difficult aspect of this challenge is obtaining good fits in the near-critical region
while observing critical constraint+matching the m*tical pressure and setting first and
second volume derivatives to zero at T=Tc and V=V,. The degree of success here affects
not only the calculation of density at T=Tc, but also afTects the equation's fit to saturation
and thermodynamic property data at subcritical temperatures near the critical point.
This work has shown that for a good fit to the critical isotherm, it is first necessary that
the regressed value of the coefficient of the v2 term provide a good prediction of the
second vLid coefficient at the critical temperature. A sensitivity analysis showed
(Figures 3-1 1(a-f)) that in the vapour-like region dong the critical isotherm of methane.
ody the V' makes a significant contniution to the predicted pressure- It follows, then,
93
that errors in pressure and specific volume dong the vapour-like region of the critical
isotherm--especially in the region leading up to the critical point-are largely due to
discrepancies between the best-fit value of the v2 term's coefficient (Bo) and the fluid's
actunl second virial coefficient. Conversely, we would expect that the closer the best-fit
value is to the actual second virial coefficient, the lower would be the error along the
vapour-like portion of the isotherm. This was verified for methane in Section 3.2.5, as
shown in Figure 3-9. In this case, where the value of Bo in Eq. (21) was set equal to BRT
(where B is the second virial coefficient of methane, calculated using the Tsonopolous
(1974) correlation), the maximum error in pressure along this part of the critical isotherm
was 0.25%. This value is lower than the maximum errors in the same region for the new
EOS (1 -2%) and the two B WR modifications (B WRS: 1 -3%; SBWR: 0.5%), all of whose
best-fit values of Bo deviated from the "correct" value. And for these three equations, the
larger the deviation fkom BRT, the higher was the error in the calculated pressure.
By contrast to the vapour-like region, four of the seven parameters in the new EOS
contribute significantly to the calcdated critical pressure: b, Bo, Co, and to a lesser extent
Do. And all parameters make significant contributions in the liquid-like region. As a
consequence, the terms that do not contribute at critical md vapour-like volumes-the
exponential term, and the V? term-have to be flexible in the liquid-like region.
Otherwise, the coefficients of the remaining terms will be forced to take on values that
compromise accuracy in the vapour-like and near-critical regions, where only they
contribute. For example, if the accuracy of the equation's fit in the liquid-like region
requires the coefficient Bo to take on a vdue different fiom BRT, then accuracy in the
94
vapour-like region will suffer. Lf, however, the exponential and higher-order terms are
flexible and can accommodate a wide range ofBo values while still maintaining the same
accuracy in the liquid-like region, the Bo term will be less impoamt here, and Bo will
tend toward the value that improves accuracy in the vapour-like region-a region where
accuracy depends solely on the value of Bo, since it's the only term that contributes there.
With these considerations in mind, it can be seen Eom Figures 4-l(a,b) to 4-7(a,b) that
the new EOS achieves a good balance between accuracy in the vapour-like, near-critical,
and liquid-like regions of pure-component critical isotherms. It gave lower average
errors in pressure than the BWRS equation for five of the seven substances, and lower
average errors in specific volume for six of the substances. It also gave lower average
errors in both pressure and specific volume than the Soave-B WR equation for three of the
seven substances. The figures show in most cases that improving the fits in the vapour-
like regions of the isotherms, especially near the critical point, would likely make the
largest reductions in the overaII average errors. And as dacribed above, such
improvements would only come tiom improving the EOS's prediction of the second
virial coefficient at T=Tc.
One way to achieve this might be to reconsider forcing the coefficient Bo to take on the
correct value corresponding to the substance's second virid coefficient at T=Tc.
Although this approach was tried briefly for the new EOS, and gave poor results in the
Liquid-like region of methane (Figure 3-9), this was likely because the exponential term-
which was essentially an attempt to fit the m r between the smoothed experimental data
and the sum of the low-degree poIynomial terms in the EOS-was not specificalIy
95
designed for this case. It is quite possible that this problem could be overcome, and
overall errors in pressure and volume could be reduced along the critical isotherm, by
modifying the form of the exponential term slightly-perhaps by re-including some
higher-order volume terms that were dropped after performing a sensitivity analysis.
The regression procedure used to fit the critical isotherms yielded the variance-
covariance matrix for the parameters being regressed. Normalizing this matrix gives the
correlation mapi t , which indicates the strength of the cross correlation between the
regressed parameters. In some cases, examination of these cross correlations can be
useful---even preferable to the sensitivity analysis performed here-for determining
whether an equation has more parameters than it needs to fit the data. However, in this
work, even though the correlation matrix was examined, the sensitivity analysis was
preferred for three reasons:
The sensitivity analysis showed the range of reduced volume where each
parameter makes its most significant contribution to the calculated pressure. This
cannot be ascertained from the correlation matrix.
The sensitivity analysis can yield information about ail parameters in the
equation, including those whose values are fixed by the critical constrakts. The
correlation matrix only shows cross correlation between the regressed parameters.
• Examination of the correlation matrix showed strong cross cornlatiom (greater
than 0.9) between the regressed parameters in Eq. (22a)-b, b, Fo, and Go. But
this is not unexpected, since these parameters all make their most significant
96
contributions over a similar range of volumes-the liquid-like portion of the
critical isotherm. And each of these parameters was included in the EOS for a
specific reason: b to avoid spurious volume roots, Eo to avoid the 'k-around"
problem at high densities, and Fo and Go as the minimum number of terms
required inside the exponential to fit the observed behavior. For this reason, even
strong cross correlation does not indicate overparameterization of the EOS in this
case.
5.2 The Covolume Parameter
As detailed in Section 3.3.5, a van der Waals-type covolume was adopted in the first term
of the new EOS to solve the problem of spurious root formation, at least for the three
substances examined. Prior to being included in the equation on these grounds, the
covolume had been examined as one of three alternatives for representing repulsive
intermolecular forces within the equation. It was shown that incorporating a covolume
into the first term not only solved the spurious root problem, but also improved the
equation's fit to the critical isotherm of methane.
Part of the appeal of adding a covolurne to the equation is that without some
representation of repulsive forces, the EOS has a zero volume at infinite pressure.
Although the new EOS is empirical, and it could perhaps be argued that its behavior at
extremely high pressures is unimportant, it seems that some kind of non-zero density
limit, however approximate, improves the equation's d 'bi l i ty as a model of fluid
behavior.
97
It can also be seen firom Figures 3-7 and 3-8 that incorporating a hard-sphere term [Eq.
(9)] instead of a simple van der Wads term, would make very Little difference to the
quality of the fit to the critical isotherm. In fact, while the %AAD in specific voolume for
the hard-sphere version of the new EOS [Eq. (22c)l was slightly less than for the h t -
term covolume version [Eq. (22a)], the %AAD in pressure was slightly higher. Unless a
simple repulsive term was developed that had a strong theoretical basis, and showed
significant improvements toward fitting pure-component critical isotherms, there is no
reason at the moment to adopt a more complicated expression than the simple van der
Waals term.
5.3 Temperature Dependence
The procedure used here for developing hctions to represent the temperahue
dependence of the EOS parameters is relatively straightforward: calculate values of three
EOS parameters at several sub-critical temperatures by forcing the EOS to match
saturation pressure and saturated liquid and vapour specific volumes; at supercritical
temperatures, fit the EOS to PV data along individual isotherms making o d y the chosen
three EOS parameters adjustable; then, by trial and error, develop algebraic functions that
fit the calculated values of the EOS parameters as functions of temperature.
While the process itself may seem simple, the problem is actudy trying to fit the
calculated curves of the EOS parameters as hctioas of temperature. The difficulty
arises because of the unusual shape these curves take on near the critical point, as shown
in Figures 3-I2(a-c). The figures show that the EOS parameters tum upward suddenly as
98
temperature is increased from subcritical values. This behavior is apparently Limited to
the near-critical region only, since the parameter values shown in the figures at
supercritical temperatures appear to line up smoothly with the values at subcritical
temperatures away fiom the critical point. The difficdty this behavior creates is that in
order to preserve the exact prediction of the cridcai point and enforcement of van der
Waals conditions, any temperature function used to fit the curves in Figures 3-12(a-c)
must pass through the value at the critical temperature. And as shown in Figure 5-1, a
magnification of Figure 3-IZ(b) near the critical temperature, passing through this point
comes at the expense of accuracy in the near-critical region.
0.8 0 -7 0.8 0.9 I .O 1 .I t -2
Reduced Temperature
i 0 Calculated Values
t t
Figore 5-1 E.upanded view of the fit of Eq. (29) to calculated values of Cu(T) near the critical temperature for methane
Regardless of the form of the temperature fbnctions used to fit these curves, the same
problem wiII arise, unIess the basic EOS is altered in some way to reduce or eliminate the
99
anomalous near-critical behavior of the EOS parameters. One way may be to let more
than three EOS parameten vary with temperature. Another way may be to incorporate an
extra term into Eq. (Ua) that becomes active only at the critical point and in its
immediate vicinity. In theory, such a term could relieve some of the burden on
parameters Bo, Co, and Do that now arises from forcing the EOS to match the critical
point and meet critical conditions. If in doing so, a near-critical term could smooth out
the curves shown in Figures 3-12(a-c) and 5-1, it would be much easier to fit the
temperature dependence of the EOS parameters, perhaps with simpler functions requiring
fewer parameters. We would consequently expect an improvement in prediction of
properties such as saturated tiquid and vapour specific volumes, and enthalpy, near the
critical point.
5.4 Further Development
In absolute terms, the new EOS fits pure-component properties very well. Its
performance is comparable to and often better than established non-cubic EOS such as
the BWRS equation and the Soave-BWR equation. However, on average, the Soave-
BWR equation fits pure-component properties better than the new equation. As wen, the
new EOS has 17 parameters in total, including the covolume--five more than the
modified B WRS equation used here, and four more than the Soave-BWR equation.
Despite these concerns, the new EOS shows enough promise to warrant further
development This development should focus on three areas:
100
a Improving the prediction of the second virial coefficient at the critical
temperature, perhaps by forcing the EOS to take on the correct value.
Improving the equation's fit near the critical point, possibly by adding a term that
becomes active only in the vicinity of the critical point.
a Reducing the number of parameters used to express temperature dependence of
the EOS parameters.
Improvements in these areas would make the new EOS an excellent candidate for
generalization and extension to fluid mixtures. It must be considered, however, that with
a non-cubic equation of state, the adjustable parameten may be strongly correlated. Such
cross correlation can lead to non-unique regressed values for the parameters, making it
difficult to develop generalizations and mixing rules. In this regard, the
recommendations described aboveimproving prediction of critical second virial
coefficients, and reducing the number of adjustable parameters-may help to make
generalization easier.
6.0 CONCLUSIONS
From this work, the following conclusions can be dram:
1. The method used to develop the new non-cubic EOS-adding a low-degree
polynomial reference term to an exponential term to fit the error between the
reference term and experimental (or smoothed) PV data dong a pure-component
critical isotherm-works well for fitting critical isotherms accurately.
3 . When a low-degree polynomial reference term is fit to the vapour-iike portion of
a pure-component critical isotherm including the critical point, the error along the
isotherm between the data and the reference term is confined to the liquid-like
region of the isotherm. If the error is then plotted on semilogarithmic coordinates
(Y-axis on log scale) as a Function of specific volume, the resulting curve is
closely approximated by a straight line, meaning that a linear function of volume
can be placed inside the exponential error term.
3. The Levenberg-Marquart method of non-linear regression provides an efficient
means of fitting a non-cubic EOS containing non-hear terms to a pure-
component critical isotherm. This applies not only to the new EOS, but to the two
moditications of the BWR EOS that were investigated.
4. Using an algebraic method within the non-hear regression to enforce critical
pressure and van der WaaIs conditions at the criticd point, and solving the
resulting equations simultaneously for values of some of the EOS parameters, is
102
simpler and more reliable than using penalty fimctions combined with large
weighting factors.
5. Simultaneously minimizing errors in both pressure and specific volume along the
critical isotherm during EOS regression improves the equation's specific volume
predictions near the critical point. Using simultaneous regression instead of
regression based on pressure done makes only a small difference in the actual
numerical values of the regressed EOS parameters.
6. Considering that the exponential term in the new EOS has a finite value when
specific volume is zero, it is necessary for the reference term to tend to positive
infinity as specific volume tends to zero (or to a value of b when the EOS includes
a covolurne parameter). Otherwise, the new EOS will not exhibit the proper high-
density limit, tending instead toward negative infinity as specific volume reaches
its minimum value. This can be achieved in two ways: by incorporating a
covolume parameter into the tirst term of the equation, or by ensuring that the
coefficient of the highest-degree term in the reference polynomial has a positive
value. This observation also applies to BWR-type equations, whose exponential
term tends to zero at low specific volumes.
7. Modifjmg the EOS's representation of repulsive intermoIecular forces-either
through a van der Wads-type covolume, or a hard-sphere term-improves the
equation's fit to specfic volumes dong the critical isotherm of methane.
103
A greater improvement was observed when a covolume was placed in each
polynomial term than when it was placed in only the first term, or when a hard-
sphere term was used. However, the resulting best-fit covolume was an order of
magnitude lower than the values that resulted for the other two methods, which
had the expected order of mapmde.
8. lnciuding a Scott hard-sphere term in the EOS made almost no difference to the
fit along the critical isotherm of methane, when compared to the version of the
EOS with a covolume in the first term.
9. For the current form of the new EOS, having a linear h c t i o n of specific volume
inside the exponential term, forcing the EOS to take on the correct value of the
second virid coefficient improved the fit along the vapour-like portion of the
critical isotherm, but degraded the fit severely along the liquid-like portion.
Forcing the EOS to take on a zero value for the third derivative of pressure with
respect to volume at the critical point produced similar results.
10. Along the vapour-Eke portion of the critical isotherm, only the term with
coefficient Bo contributes significantly to the calculated pressure. At the critical
point, three polynomial terms-Bo, Co, and D-as well as the covolurne 6, make
significant contributions to the pressure. All terms in the EOS contribute
significantly dong the liquid-like portion of the isotherm.
11. When three parameters are selected to vary with temperature, plots of the
parameters as fimctions of temperature behave anomalously near the criticai
104
point, turning upward sharply over a small range of temperature. It is difticult to
match the required values at the critical temperature using analytical temperature
hnctions, and still achieve a good fit at near-critical temperatures.
1 2. Analytical temperature functions expressing the temperature dependence of EOS
parameters in terms of reciprocal temperature generally give better fits than
hct ions written in terms of temperature directly.
13. Spurious inflections and volume roots can form dong subcritical and supercritical
isotherms predicted by non-cubic equations of state. Incorporating a covolume
parameter into the first term of the EOS eliminated the problem OF spurious
volume roots at subcritical temperatures for the three substances examined in this
work. Spurious volume roots at supercritical temperatures were encountered
when EOS parameters were regressed using saturation data only. These roots
were prevented fkom forming when supercritical PVT data was included in the
EOS parameter fining procedure along with the saturation data.
15. The Levenberg-Marquart method for non-linear regression was also robust and
reliable for the overall fitting of EOS parameters directly to the database of
saturation data and supercritical PVT data.
16. The new EOS generally performs better for fitting pure component properties
than the 12-constant version of the B W equation investigated here, but only
performed better than the Soave-BWR equation in select instances. Further work
should focus on improving the new EOS's prediction of criticd second viriaI
205
coefficients, and on mhimhhg the anomalous behavior of the values of the EOS
constants at the critical point, perhaps by developing a special term that becomes
active only in the near-critical region.
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APPENDLX: EOS Constants for Selected Pure Components
The EOS constants shown in the tables are for pressure in atmospheres, temperature in
degrees Kelvin, and specific volume in cm3/mol:
Tabie A-1 Pure-component critical constants for new EOS pq. (22a)l - -
Constant Value Methane Propane n-Pentane Hydrogen Cubon Sulfirr Water
Dioxide Dioxide Tc (K) 190.56 369.85 469.77 32.98 304.21 430.65 647.29 b 22. I39 66.186 107.132 f 6.473 0 43.3 12 0 Bo -22443 19 -95585 18 -19389593 -2429 14 -2863029 -7098559 -4562 I48 cu 75 179745 392796259 1548256033 4294569 94521554 229082860 1424 13078 Do -8.589E+12 -3.755E-t-14 -4,184E+15 -9,828E+IO -8.552E+12 -6.495E+13 -3,738E+12 61 2.329Ei-14 -2.227E+ 16 -6.322.E+17 ?.214E+ll 1.116E+14 -4.8 13E+ 14 5.596E-i-13 Fo 590868 1208320 10 14273 206186 480 876597 1067131 Gn -0.16542 -0.08 1 85 -0,05200 -0.258 1 1 -0.03206 -0.1 1993 -0.23383
Table A-2 Pure-component critical constants for Soave-B WR equation pq. (4)]
Constant Value Methane Ropane n-Pentane Hydrogen Carbon S u l f i Water
Dioxide Dioxide
Table A-3 Pure-component critical constants for BWRS equation @q. (2)J
Coastant VaIue Methane Propane n-Pentane Hydrogen Carbon S u k Water
Dioxide Dioxide B - 1902789 -7760526 -15729256 -202908 -3022 143 -5699602 -4545263 C 3305058 1 204575267 542265045 34 13955 50678667 1724352 I8 94269269 D 4.088E+12 3.246E+ 14 3.987E+15 5.671E+10 5.029E+ 12 1.8 I6E+ 13 -2,297E+10 E 504930 12 509324592 1757912940 2 146833 8 1 t 77960 16 1007423 42398522 F 7283 28362 6667 1 3901 6568 12415 3377
Table A-4 Pure-component temperature constants for new EOS [Eqs. (28)-(30)]
Constant Value Methane n-Pentane S u k
Dioxide
I l l
Table A-5 Pure-component temperature constants for Soave-B WR equation @qs. (31, c, e)]
Constant Value Methane n-Pentane S u k
Dioxide
Table A d Pure-component temperature constants for BWRS equation [Eqs. (42% b, c)]
-- . - - - -
Constant VaIue Methane n-Pentane S u I h
Dioxide